UNIVERSITY OF CALIFORNIA DIVISION OF AGRICULTURAL SCIENCES GI ANN I NI F 0 UN DAT I 0 N 0 F AG RIC ULT UR AL EC 0 N 0 MIC S

Cattle Feedlot Marketing Decisions Under Uncertainty

J. B. Bullock and S. H. Logan

Giannini Foundation Monograph Number 28 • April, 1972

CUGGB9 28 1-46 (1972)

CALIFORNIA AGRICULTURAL EXPERIMENT STATION

Uncertainty with respect to future prices, both for fat cattle and for feeder cattle, are important aspects in feedlot decision-making processes. This study illustrates, through the use of statistical de­cision theory, how such uncertainty can be considered directly in the decision process. Models are developed for determining deci­sion rules for marketing cattle currently on feed or continuing their feeding for another time period, for the procurement of feeder animals, and for planning both procurement and marketing over a six-month planning horizon. A monthly price forecasting model serves as the basis for all the decision models. The predict­ing model is a recursive system involving forecasts of marketings of fat cattle, numbers of animals on feed, and other inventory levels to provide information for the primary price forecasting equations.

THE AUTHORS:

J. B. Bullock is Assistant Professor of Agricultural Economics at North Carolina State University, Raleigh, North Carolina.

S. H. Logan is Professor of Agricultural Economics in the Experi­ment Station and on the Giannini Foundation, University of California, Davis.

/. B. Bullock and S. H. Logan

CATTLE FEEDLOT MARKETING DECISIONS UNDER UNCERTAINTY1 2

•

INTRODUCTION

RISK AND UNCERTAINTY are dominant characteristics of many agricultural processing operations. Decisions on pro­curement, production, and marketing are made on the basis of imperfect knowledge about future conditions, and the stochastic nature of such conditions may result in a financial loss for the firm despite the care taken in making deci­sions. To the extent that risk and un­certainty can be incorporated into the decision-making process, the firm can

reduce the chances of loss from unex­pected market fluctuations.

This monograph's objectives are: 1) to illustrate through a practical ap­plication the use of probability theory in management decision making and 2) to develop a set of decision criteria to assist cattle feeders in making purchas­ing and marketing decisions when faced with uncertainty about future cattle prices.

THE PROBLEM

Uncertainty about future prices and the feedlot performance of cattle are challenges to cattle feeders faced with decisions on purchasing feeder cattle or marketing slaughter animals. Unex­pected price changes, sickness, death loss, or simply low rates of gain of cattle on feed can lead to negative profits. The profitability of the cattle feeder's pur­chasing and marketing decisions hinges on his ability to anticipate future prices and to assess accurately the potential feedlot performance of the cattle on feed as well as his ability to operate efficiently the physical facilities of the feedlot.

The decision to place cattle on feed is based on a comparison of expected value of the cattle at the end of the feeding

period with the current cost of feeder cattle plus expected feeding costs. The feedlot operator, therefore, must antici­pate market conditions three to six months ahead and buy the age, quality, and type of feeder cattle that he expects will yield the most profit. Furthermore, he is faced with the possibility that his anticipations may be incorrect and ex­pected positive returns may turn out to be negative.

Although operating practices vary among California feedlots, the "aver­age" practice is to place 600-pound steers on feed with the intention of marketing them as 1,000-pound slaughter steers approximately 150 days later (Logan and King, 1966). In some instances,

1 Submitted for publication September 20, 1971. 2 This research project was developed jointly by the U.S. Department of Agriculture, Market

Economics Division, Economics Research Service, and the University of California, Davis.

[1]

2 B~illook and Logan: Cattle Feedlot Market~ng Decisions Under Uncertainty

however, the cattle may be sold for slaughter at weights ranging from 800 pounds to more than 1,200 pounds. Thus, there is a range of about 400 pounds over which the operator must exercise the decision of whether to sell a particular lot of cattle at their current weight or to continue to feed the cattle at least another time period, which, for this study, is 30 days (one month). The feed-or-sell decision is based on a com­parison of costs of feeding another 30 days with expected returns from the additional feeding. If the cattle are sold too soon, added profits may be fore­gone; however, if cattle are held on feed too long, profits can be decreased.

If the feeder can generate additional information concerning future condi­tions, he may be able to reduce the de­gree of random variation surrounding the possible outcomes of his decisions. Thus, limits may be placed on certain types of decisions as a result of such additional information. For example, feeder cattle may not be purchased un­less predicted fat cattle prices six months ahead are at a pre-specified level, or have a certain probability of being at­tained.

To provide cattle feeders with infor­mation other than merely current price relationships, this study utilizes a monthly forecasting model for slaughter and feeder cattle prices developed pre­

'

viously (Bullock, 1968). The initial phase incorporates the information pro­vided by the price forecasting model into a Baye~ian decision framework to arrive at a set of marketing strategies. These strategies can be used by the cat­tle feeder t9 evaluate the feed-or-sell alternatives, given the current weight of a particular lot of cattle and the current price of slaughter cattle. The second phase utilizes the results of the price forecasting model to develop a set of de­cision criteria for purchasing feeder cat­tle. The final phase of the study com­bines the results of the previous two phases into a six-month planning model.

Although the decisions made in these planning models aim at increased prof­its, the various interrelationships among the decisions are not considered ex­plicitly. For instance, the decision to re­tain cattle on feed rather than sell them for slaughter. precludes the use of that pen space for new, lighter-weight feeder animals. However, it is possible that given feeder cattle and fat cattle price relationships, the optimum decision would be to replace older animals with younger ones whose weight gain will be greater than that of the oider animals. These interrelationships are a separate study in themselves and are not con­sidered here; thus, the model is of a partial nature.

CATTLE FEEDING OPERATIONS

Cattle feeding in California (see Lo­gan and King, 1966, and Hopkin and Kramer, 1965) is characterized by large specialized feedlots. In 1969, 99 per cent of the fed cattle marketed in the State came from lots with capacities of 1,000 head or more. Moreover, about 82 per cent of the marketings originated in lots with capacities of 8,000 head or more. This is in sharp contrast to the

major cattle feeding regions of the Mid­west. For example, in 1969 in Iowa, lots with capacities of less than 1,000 head marketed 91 per cent of the fed cattle. Iowa lots with capacities exceeding 8,000 head accounted for less than· 2 per cent of the marketings (USDA-SRS, 1970).

The large numbers of cattle fed per feedlot in California place added em­phasis on the operator's purchasing and

3 Giannini Foundation 'Jifonograph • Number 28 • Maroh, 1972

marketing decisions. Proper timing and better accuracy qf these decisions can mean several thous~nd dollars in added revenue. Thus, in addition to achieving efficient gains for ahimals on feed, the feedlot operator must be aware of cur­rent and expected market conditions to purchase and sell c11ttle effectively.

Problems of price uncertainty

The primary som,<ce of risk in cattle feeding is imperfect knowledge about future prices. A~· .·experienced cattle feeder can estimate fairly accurately the cost of feeding a pa,.rticular lot of cattle to the desired slaughter weight and the grades they will att~in, and he can affect some of the factors of feedlot perform­ance. However, future prices are de­pendent on many interrelated variables and beyond his control. Consequently, his information about future prices is less precise.

However, some knowledge of future slaughter cattle prices is necessary for decisions about placing cattle in the feedlot and for determining the best time to sell fed cattl~. Information about future prices is probably most crucial for the purchasing .decision because a three- to six-month forecast of slaughter cattle prices is needed.

The purchase decision is based on the expected feeding margin-the price per hundredweight received for the finished animal minus the . price per hundred­weight paid for the animal entering the feedlot. The break..even margin is de­fined as the margin necessary to cover all costs of feeding .. The difference be­tween realized m~rgin and the break­even margin repres~nts the profit (or loss) per hundredweight of fed steer. The

accuracy of the feedlot operator's pro­jection of slaughter cattle prices is criti­cal. If future slaughter cattle prices are overestimated and additional cattle are fed, negative net returns may result. On the other hand, if prices are underesti­mated, positive returns may be foregone if cattle are not placed on feed.

Knowledge of future prices is also im­portant in determining when to sell fed cattle. The decision to feed another month or sell at current weight is based on a comparison of the expected margi­nal revenue from continued feeding with the cost of feeding as indicated in equa­tion (1).8

or

where

C = cost of feeding another month, W = current weight, G = weight gain,

Pi = current slaughter price, P c1+1J = price next month,

AP Pc1+1> - Pt

A particular lot is continued on feed only if C < PiG + AP (W G). As in the purchase decision, the primary source of risk in the selling decision is the price change that may occur.

Some of the uncertainty about future price movements can be removed by forward contracting for purchase and sale of cattle4 and by hedging opera­tions in the futures market. However, it is difficult for California cattle feeders to hedge effectively in the futures mar­

3 This is a simplified version of the marketing decision problem because replacement aspects are ignored. For a discussion of replacement decisions, see Faris (1960).

4 Logan and King (1966, pp. 21-23) reported that about half of the feedlots surveyed used con­tracts for purchasing feeder cattle and advance contracts (30 days) for sale of fed cattle were used for 73 per cent of cattle marketed.

4 Bullock and Logan: Gattie Feedlot Marketing Deol,swns Under Uncertainty

ket because a viable West Coast futures market does not exist and hedging op­erations must be transacted in the Mid­western market. Futures contracts have rigid specifications as to weight, grade, and location of cattle that can be de­livered under contract. Thus, to utilize the Midwestern futures market, the California cattle feeder has to adjust Midwestern cattle prices for locational differences and for the quality of cattle in his feedlot. Although slaughter cattle prices in California and the Midwest are interrelated, they are not perfectly correlated. Thus, in the short run, prices in one market may be declining while in the other market prices may be hold­ing steady or even increasing slightly.5

In such cases, price movements adverse to the California cattle feeder are mag­nified if he is using the Midwestern mar­ket for hedging operations. Conse­quently, while the futures market may reduce risk, it does not completely re­move price uncertainty for the cattle feeder.

Other sources of risk Poor feedlot performance is another

important source of risk for the cattle feeder. Scientific management practices may have helped to reduce sickness and death loss of cattle on feed. Veterinari­ans and nutrition experts frequently are employed by large feedlots to re­duce these risks, but they have not been eliminated.

Typically, the cattle feeder operates on a narrow margin of profit, basing his purchase decision on what he thinks the cost per pound of gain will be for the feeder cattle and their expected value at the end of the feeding period. If the feeder cattle do not gain as efficiently as he had anticipated, or if feed prices rise unexpectedly, added cost per pound of gain may eliminate expected profits, re­gardless of the accuracy of his price expectations. Similarly, if the cattle do not reach the planned slaughter grades, their value at the end of the feeding period will be less than expected and negative profits may result.

THE DECISION MODEL

The problem of decision making under uncertainty can be characterized as a decision maker faced with choosing the optimal course of action, A,., from a set of m possible actions. The outcomes of th~se various actions are dependent on the occurrence of alternative states of nature E>ii j = 1, 2, ... , n. The states of nature are values of an exogenous factor that directly affects the outcome of a particular action but is beyond the con­trol of the decision maker; at least, this factor cannot be controlled with cer­tainty. For example, if the set of actions represents different rates of fertilizer

applications for corn, the states of na­ture might be alternative levels of rain­fall. Thus, for each possible action Ai, A2, ... , Am, there are n potential out­comes, one for each state of nature. Each outcome, Ai;, can be represented as a point in an action-state plane, A.;1 = (Ai, E>;). The matrix formulation of the outcome plane is presented in table 1.

For example, the outcome (profits) of a decision to feed two types of steers (low quality and high quality) will de­pend on the prices of slaughter cattle at the end of the feeding period. Thus, E>1

6 Divergent movements are limited by the amount of transportation costs between the two markets because intermarket shipments become profitable if prices differ by more than transfer costs. However 1 price movements within this range could exceed feeding margins in some cases.

5 Giannini Foundation Monograph • Number 28 • March, 1972

TABLE 1

MATRIX REPRESENTATION OF OUTCOME PLANE

States of nature

Action 01 02 01 a.

Ai ................ lln l\1; Alo

A2................ ;1.,, "" "•; "•· ........ ........ "" . .. ... ....... A,: ........ AH ).,;<,i Aii..... "'" ., .... ,,,,,,,,, .. .······ Am................ Am! Am2 Ami Amo

may represent high slaughter-cattle value of e occurs (cost per pound of gain prices; e2, average prices; and ea, low is assumed to be known with certainty prices. The outcome of decisions A1 in both cases). This decision problem is (feed high-quality steers) and A2 (feed then as follows: low-quality steers) will depend on which

Action

Ai (feed .high-quality steers) A2 (feed low-quality steers)

States of nature

el e2 es (high prices) (average prices) (low prices)

where A12 is the profit per head from feeding high-quality steers when aver­age prices are received at the end of the feeding period.

To make rational and consistent de­cisions about the action-state-outcome combinations, a utility index or some sort of preference ordering must be as­signed to the set of outcomes. If the decision maker's preferences among the outcomes are consistent with von Neu­mann-Morgenstern utility axioms (see also Luce and Raiffa, 1965, pp. 23-31) it is possible to define a utility function, UiJ = u(")l.;1), that will map the outcomes into a utility plane.

Von Neumann and Morgenstern (1947) show that if:

1. the individual has a complete and transitive preference ordering over the set of all possible prospects, that is, (a) for any two prospects u and v, one and only one of the following relations holds:

u v, u > v, u < v6

(b) u > v, v > w implies u > w 2. u < w < v implies the existence of

an a(u) + (1 - a)v < w, and u > w > v implies the existence of an a(u) + (1 - a)v > w, where

6 Where: =implies indifference between prospects, > is read as "is preferred to," and < is read as "is not preferred to."

6 Bullook anrJ, Logan: Cattle FeerJlot Marketing DelYiSwns Unaer Unoertainty

TABLE 2

MATRIX FORMULATION OF DECISION PROBLEM UNDER

States of nature

Action 01 02 0; en

Ai. .. ....... uu Ul2 U!j Uln

A•. ... .. ............. ... '"' U22 U!j U2n

.... .. .. ...... .. .... .... ..... A;. .. .. ....... .... .... ... .. .. .. .. ..

un UiZ Uij Uin

... ... .... Am.. .... .. .. Um! Um2 um; Umn

0 <a< I, and 3. if it is irrelevant whether a combi­

nation of two prospects is obtained in two successive steps-first the probabilities a, I - a, then the probabilities (3, 1 - {3; or in one operation with the probabilities 'Yi

1 - 'Y where 'Y = a{3 (that is, com­plex choices can be partitioned into simpler choices to facilitate evalu­ating preferences)

au (1 - a)v = (1 - a)v + au

and

a[{3u (1 - f3)vJ + (1 - a)v ='YU (I - 'Y)V

then there exists a utility function u on the set of prospects.

In other words, for each prospect P; there exists a number Ui u(P.) which is called the utility of Pi. This function has the following properties (Chernoff · and Moses, 1959):

(a) u(v) > u(w) if and only if the individual prefers v to w.

(b) If P,. is a prospect of receiving v with probability a or w with probability (1 - a) then u(P,,) = au(v) + (1 - a) u(w).

As a matter of practical application, it is usually assumed that the utility

function is linear with respect to money over the relevant range. Consequently, maximization of monetary gain is equiv­alent to maximizing utility.

Thus, the decision problem can be seen as stated in table 2. Given a set of possible actions, A, the set of alternative states of nature, e, and the utility index Ui;, associated with the selection of ac­tion A; and the occurrence of 01 (out­come f..;1), select the action that is in some sense optimal-where optimality is defined by the particular decision criterion used. Various decision criteria are available, many of which deal with decisions with no knowledge at all about the states of nature.

However, most of these decision cri­teria have serious shortcomings as dis­cussed by Luce and Raiffa (1965, pp. 278-286). See also Chernoff (1954) and Radner and Marschak (1954).

Bayesian decision theory

Few decision problems fall into the category of complete uncertainty, i.e., where the decision maker has no knowl­edge of the likelihood or distribution of e. Given the volume of public and pri­vate information currently available, some a priori information regarding the relative frequency of e in the past generally can he . obtained. Thus, em­phasis in decision theory has shifted to

7 Giannini Foundation Monograph • Number 28 • MIM·oh, 1972

the estimation of Bayesian strategies ;7

i.e., the selection of optimal actions based on some a priori information, either objective or subjective, about the probability distribution of the states of nature, P(0).

The Bayesian approach to decision making can be stated as follows: Given a set of m possible actions, the set of n alternative states of nature, and the utility index associated with each out­come, along with a vector of a priori in­formation about the relative frequency of 0,

where P(01) is the a P(0) = P(01) priori probability that

state 01 will occur

select the action A; for which expected utility u, = 4 u.1P(01) is a maximum.

j

The a priori information can be any information that the decision maker has about the relative frequency of 0. This information is expressed in the form of a probability distribution P (0) that pro­vides some indication of the likelihood of a particular value of 0 (state of na­ture) occurring. It may be nothing more than a subjective evaluation of the probabilities by the decision maker, or it may be derived mathematically from data on the relative frequency of 0 in the past.~

In addition to the a priori knowledge of the probability distribution, P(0), it may be possible for the decision maker to gain additional information about the likelihood of a particular state 01 by performing an experiment Z (with re­sults Zk, k 1, 2, ... , n) that serves as a predictor of 0.8 That is, it may be possible to construct a conditional prob­ability distribution, P(0IZ), which in­corporates the a priori information, P(0), with information about the past performance of z as a predictor of 0. The a posteriori probability distribution, P(EllZ), can be calculated using Bayes' Formula:9

P(EljZ) = P(Zj0)(P0) P(Z)

The experimental information ex­pands our knowledge about the likeli­hood of 0 from the P(El) vector to an (nxn) matrix of conditional probabili­ties (table3), whereP(0JIZk) is the proba­bility of 01 occurring given Zk as the experimental result (prediction of 0). If the experiment Z is a perfect predictor of 0, table 3 will consist of ones along the diagonal and zeros elsewhere.

With data provided by the experi­ment, the Bayesian strategy becomes: Given a projection of 0 (for example, Zk) select the action Ai for which the expected utility

u: 4 Ui1P(El1!Zk) (3) j

is a maximum. Thus, the Bayesian

1 (Jeffery, 1965); (Raiffa and Schlaifer, 1961); (Weiss, 1961); (Luce and Raiffa, 1965) and (Chernoff and Moses, 1959).

8 The experiment, Z, can be anything that is used as an estimator of 0. It may consist of simply observing the current state of nature 0; and assuming that the value of 0 at the time of payoff will also be E>;. The price forecasting model developed in the following section functions as the experiment for this study.

9 For a derivation of Bayes' Formula, see Hoel (1962, p. 16). This procedure is used to calcu­late a posteriori probability distributions in this study. For other applications see Eidman et al. (1968) and Dean et al.. (1966). Depending on the nature of the experimental data, it may also be possible to estimate P(0IZ) directly without the use of Bayes' Formula.

8 Bullock and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

TABLE 3 MATRIX OF A POSTERIORI INFORMATION

Experimental results

States

01. ...... ' ... .. 02 ..............

....... ............................. , ........ @;.............. ....... ..... ,,,................ e ...............

Z1 z,

P(El1!Z1) P(El1JZ2} P(02IZ1) P(El2IZ2)

P(0;!Z1) P(01IZ2l

P(Eln!Z1) P(0.!Z2)

strategy consists of a set of optimal ac-. tions, at least one for each experimental result.10

Value of the data

The derivation of Bayesian decisions by using only the a priori probability distribution P(9) is referred to as the "no data" problem. Decision problems using a posteriori distributions are called "data" problems. The difference in ex­pected incomes resulting from using the "data" strategy bundle relative to the "no data" strategy can be interpreted as the value of the data, i.e., the value of the information provided by the ex­periment.

The expected value of the "no data" strategy is defined above as ai =

}; ui;P(91). The expected value of fol'­;

z.

P(01IZ•l P(02IZ•l

z.

P(01IZ.) P(02!Z.)

P(0;IZ•l

P(0n!Z•l

lowing the ''data" strategy is calculated by multiplying the expected value of the optimum action for each experimental result by the probability of observing the appropriate experimental result, P(Z), and summing over all possible re­sults

}; [}; UijP(0;1Zk)JP(Zk) (4) " j

The expression in brackets was defined in equation (3) as u~ (expected utility of action Ai given Zk as a prediction of 9). Thus, the above expression reduces to }; ~P(Z"). Therefore, the value of the k

data is defined as

V =}; u~P(Z") - u;. k

MODEL FORMULATION Within this general framework of de­

cision theory, four models are set up as a framework for analysis. Models I and II are short-run models and deal only with marketing decisions; Model III involves longer-run purchasing decisions; and Model IV combines marketing and purchasing decisions for a six-month planning horizon.

Model I is a direct application of Bayesian decision theory to the. prob­lem of feedlot marketing decisions. It is designed to determine the minimum ex­pected price change required to induce feeding a particular lot of cattle another month, given the current weight of the cattle and current slaughter cattle prices. The model incorporates information

lQ It is possible that two or more actions could have the same expected utility for a. given experimental result ..

9 Giannini Foundation Monograph • Number 28 • March, 1972

about the cost of the additional gain and expected slaughter grade of the cattle 30 days hence with a posteriori informa­tion (in the form of probability distribu­tions) about the accuracy of the price forecasting model to arrive at a set of feed-or-sell decision rules.

Model II is an extension of Model I. For animals weighing less than 1,000 pounds, it is not unreasonable to con­sider extending the feeding period an­other 60 days. Furthermore, it is con­ceivable that a sell decision could be generated by Model I when a one­month price projection is considered but that it might be profitable to continue feeding the animals if we consider ex­pected prices 60 days hence. Model II, therefore, is constructed to evaluate the feed-or-sell decision based on 60-day price projections. This model is appli­cable only if (a) current weight of the cattle is less than 1,000 pounds and (b) a sell decision arises in Model I.

Model III develops a set of buy-or­not-buy decision criteria for feeder cat­tle based on expected feeding margins. Estimates of cost per pound of gain and proportion of cattle feeding to choice grade are combined with projected slaughter cattle prices to determine ex­pected feeding margins.

Model IV, a six-month planning model, incorporates the decision rules developed in the first three models into a simulation model. Model IV simulates the buying, feeding, and selling activities six months into the future, given the capacity of the feedlot, current inven­tories of cattle on feed by weights, and projected feeder and slaughter cattle prices. This information should be help­ful to the feedlot operator in making forward arrangements for financing, feed acquisition, and contracting for pur­chase of feeder cattle and/or sale of slaughter cattle.

DATA REQUIREMENTS

Model IV requires the same data as the first three models plus longer-run projections of prices; therefore, a discus­sion of data needs for this model auto­matically covers the needs of the first three models. To make tentative deci­sions about purchases and sales six months in advance, feeder cattle prices must be projected six months into the future and slaughter cattle prices 11 months ahead. For example, a tentative decision regarding placements i:;ix months ahead requires a six-month projection of feeder cattle prices plus an estimate of slaughter cattle prices five months later, at the end of the proposed feeding period (i.e., 11 months in advance of the planning date).

Two additional sets of information are required to develop strategies for the marketing and purchase decisions: (a)

cost per pound of gain as the weight of the animal increases and (b) the propor­tion of fed cattle that can be expected to grade · Choice or better at alternative slaughter weights. Aside from price changes, these are the primary variables in the marketing and purchase de­cisions.

Cost per pound of gain increases as weight of the animal increases because a larger proportion of feed intake is re­quired just for maintenance at greater weights (National Academy of Sciences, NRC, 1963; Garrett.et al., 1959). Al­most twice as much feed is required per pound of gain for 1,200-pound steers as for 600-pound steers. Thus, in some in­stances, feeding to heavier weights may not be feasible because the cost per pound of gain may exceed slaughter cattle prices.

10 Bullock amd Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

This r1smg cost per pound of gain, however, may be offset to some extent as additional Good grade steers attain Choice grade, because the proportion of slaughter steers grading Choice in­creases (and thus their value increases), ceteris paribus, as weight increases.

The input requirements, then, needed to develop the models formulated above can be summarized as follows:

1. A monthly price forecasting model to project (a) slaughter cattle prices 11

months ahead and

(b) feeder cattle prices six months ahead.

2. A posteriori probability distribu­tion of price changes, given projec­tions of the price forecasting model.

3. Data relating the cost per pound of gain to weight of steer.

4. Data relating proportion of cattle grading Choice to slaughter weight.

In addition, a probability distribution of price changes in the past will be used as the basis for a "no data" strategy with which to compare the results of "data" strategy utilizing the price fore­casting model.

APPLICABILITY AND GENERAL SPECIFICATIONS

The decision rules developed in this study are based on typical cost and pro­duction relationships of California feed­lots. Because not all California feedlots have the same cost structure or follow the same operating procedures, the question arises how applicable decision rules based on average relationships are to specific problems faced by an indi­vidual feedlot operator.

The applicability of the decision rules to a wide range of decision problems de­pends on how sensitive the models (used to derive the rules) are to the above mentioned variables. Do slight changes in cost relationships or variations in feed prices give rise to a different set of decision rules? A sensitivity analysis of tl/e models (explained in detail later) indicates that the same decision rules would be derived for a range of feeding costs. Thus, the rules developed from "typical" or average cost and produc­tion relationships should have a rather

general application to decision prob­lems faced by California feedlot opera­tors.

The "typical" feedlot is assumed to purchase 600-pound Good grade feeder steers at prices reflected by the Stock­ton, California, market and to sell Good and Choice grade slaughter steers at prices indicated by the El Centro, Cali­fornia, market. El Centro and Stockton were selected as the representative mar­kets for this study because they are important markets in the State and time series of price data are available for use in developing the price forecasting model. Their selection. is not a limiting factor, however, because prices of feeder cattle and slaughter cattle throughout the State are affected by the same factors. Prices between geographic points in the State are interdependent and price changes from one time period to the next (the basis for this study) will be essentially the same for each point.

PRICE FORECASTING MODEL

A brief summary of the monthly price been revised with more recent data forecasting model, developed in detail than those used in Bullock (1968). Es­elsewhere, is given here. However, the sentially, the model is recursive in na­equations used in this application have ture with prices being related to pre­

11 Giannini Foundation Monograph • Number 28 • March, 1972

dieted values of other variables. The model predicts monthly prices of 900­to 1,100-pound Choice grade slaughter steers at El Centro as a function of lagged prices and predicted marketings of fed cattle in California, Arizona, Texas, Colorado, and the North Central region, based on the period 1960-69. Marketings, in turn, depend on the num­ber of cattle and calves on feed at the

· beginning of a particular quarter. The latter variable requires prediction, also, when prices are to be forecast further ahead than the current quarter.

The basic structure of the price fore­casting model is outlined by the follow­ing equations:

Price Forecasting Equation (PJi)

(Choice grade slaughter cattle prices are predicted as a function of pro­jected marketings of fed cattle in various regions, lagged prices of Choice grade steers, and quarterly dummy variables.)

Fed Cattle Marketings (MJk)

(Fed cattle marketings in region k are projected as a function of pre­dicted or actua,l cattle on feed by weight group in the region plus quarterly variables and a linear time trend.)

....... Cattle on Feed Projections (W:hk)

(A projection n quarters ahead of cattle on feed in weight group h for region k is a function of January 1

inventories of steers and calves and cattle on feed by weight group (ex­cluding :2: 1,100 pounds) in region k.)

Steer Inventory (S<t+1Jk)

(Steer inventory on January 1 for the coming year is a function of January 1 inventory of calves for the current year in the region, the January 1 in­ventory of beef cows for the previous year, and the average Kansas City­Chicago feeding margin for the cur­rent year up to the time of projection.)

A

Calf Inventory (C<t+iJk)

..... C<t+1Jk = hk(PP'<t-iJ, BC1k, BH1k, PP') (10)

(Calf inventory on January 1 for the coming year is a function of the aver­age price of feeder steers at Kansas City the preceding year, inventories of beef cows and beef heifers on Jan­uary 1 of the current year in region k, and the average price of feeder steers at Kansas City for the current year up to the time of projection.) The symbols used in the above func­tions are as follows:

BCtk =January finventory of beef cows for current year ·in region k (1,000 head).

BC<t-iJk = January 1 inventory of beef cows for previous year m region k (1,000 head).

Ctk =January 1 inventory of calves for current year m region k (1,000 head).

" Cc1+1Jk = projection of January 1 in­ventory of calves less than one year old for the coming year in region k (1,000 head).

h = 1, 2, 3, 4, where 1 = 500 to 699 pounds

12 Bu1fool~ and Logan: CatUe Feedfot Marketing Decisions Under Uncertainty

2 = 700 to 899 pounds 3 = 900 to 1,099 pounds 4 = more than 1,100

pounds. i = the length (in months) of

the projection being made. j = the month of the quarter

and equals 1, 2, or 3. k = the feeding region. n = the length (in quarters) of

projections of cattle on feed. M = average Kansas City-Chi­

cago feeding margin for the current year up to the time of projection (dollars per hundredweight).

M1,. = projected marketings of fed cattle for the jth month in region k (1,000 head).

PP'= average price of feeder steers (all weigh ts and grades) at Kansas City for current year up to time of projection (dollars per hun­dredweight).

PP' (1-1) = average price of feeder steers · (all weights and grades) at

Kansas City in preceding year (dollars per hundred­weight).

Pi• = projected price for the jth month i months ahead (dol­lars per hundredweight).

Pvil-1 =projected price for month preceding the (ji) projec­

j tion. If i = 1, i.e., a one month projection, the ac­tual lagged price is used (dollars per hundredweight).

P 0•/-12 = price 12 months previous to the month for which price is being projected (dollars per hundredweight).

Qi, Q2, Qa = quarterly dummy variables. sk =January 1 inventory of

steers (1,000 head) one year old and older in region k or its major supply region.

S<1+1Jk = projection of January 1 in­ventory of steers for the coming year in region k (1,000 head).

t = current year. T a linear time trend = 0 for

the first quarter of 1960 and increases by 1 each quarter. T 36 for the first quarter of 1969.

whk = cattle on feed in weight group h (h = 1, 2, 3, 4) in region k at the beginning of either the current quarter or total for most recent two quarters (excluding h = 4).

...., a Wnhk an n-quarter projection of

cattle on feed (1,000 head) in weight group h in region k. a refers to the quarter from which the projection is made and equals 1, 2, 3, 4 ..

The model is subdivided by quarters because cattle and calves on feed data are available only by quarters. Market­ings are forecast with separate equations for the first, second, and third months of the quarter. Prices, in turn, are also forecast with a separate equation for each month of a quarter and with a separate equation depending on the num­ber of months ahead for which the pre­diction is being made. In this sense, there is one equation for a four-month projection of prices in the first month of a quarter, another equation for a four-month projection of the second month of the quarter, and so forth. The coefficients of price predicting equations are given in table 4.

The use of separate equations for each type of forecast precludes the use of the standard error of the estimate in deriving the needed probability distri­butions for the decision problem, as will be discussed later. However, the standard errors of the estimate are given in table 4.

TABLE 4

PRICE FORECASTING EQUATIONS FOR CHOICE GRADE, 900-1,100 POUND SLAUGHTER STEERS, EL CENTRO, CALIFORNIA

Lagged prices Projected markel:Jngs of fed cattle Quarterly dummy variables• 'Constant

Equation Lengthof price

projection

Month of

quarter

term Pct-1) P<t-t>l California. Arizona. Texas Colorado North

central Q, Q, Qa R' Sy y

a bi b2 ba b< ba bo b1 bs bo bin

months dollars/cwt

1...... ' .. ' 1 1 11.5938 .7115 .0846 -.0122 -.0253 -.0069 .0606 -.0075 .9529 -.4535 .1666 .9206 ,6692 25.59 (.0669)t (.0886) (.0134) (.0496) (.0099) (.0245) (.0040) ( .6426) (. 7124) ( .4935)

2.......... I 2 28.3505 .8818 -.0900 - .0147 -.0807 .0053 .1051 -.0260 -.0338 -.6440 .2656 .9421 .6477 25.61 (.0837) (.0786) (.0115) (.0376) (.0124) (.0299) (.0063) ( .3772) (. 7741) (.2105)

3 .......... 1 3 10.2923 .8583 - .1552 -.0237 .0070 -.0105 .0611 -.0034 -.3786 - .5616 .1421 .9182 .7860 25.83 ( .0008) (.0917) (.0136) (.0440) ( .0133) (.0377) ( .0049) (.3542) ( .6934) (.4218)

4 .......... 2 1 - 2.5770 .6490 - .1199 .0217 .0198 .0303 -.0995 .0148 .4833 -.0155 .5964 .8307 .9775 25.59 (. 1162) (.1705) (.0213) (.0654) (.0164) (.0406) (.0136) (1.6437) (1.4582) (.9633)

5 .......... 2 2 34.1724 .7553 -.0375 - .0201 -.0776 .0001 .1364 -.0313 .1128 -1.1682 .4036 .8500 1.0587 25.42 (.1620) (.1310) (.0193) (.0634) (.0217) (.0500) ( .0109) (.6392) (1.3083) (.3481)

6.......... 2 3 11.7481 .8143 -.1735 -.0258 .0077 -.0116 .0655 -.0032 -.4066 -.6332 -.0876 .8314 1.1287 25.83 (.1527) (.1363) ( .0197) (.0640) (.0193) (.0558) (.0075) (.5117) (1.0118) (.6075)

7.......... 3 1 1.1316 .7104 .1935 .0231 -.0315 .0394 -.1056 .0136 -.0229 .4027 .9101 .8097 1.0362 25.59 (.1430) (.1815) (.0226) (.0742) (.0170) (.0430) ( .0144) (1. 7421) (1.5612) ( .0161)

8.......... 3 2 20.6716 1.3981 - .0631 -.0867 -.2037 -.0312 .1513 -.0148 -2.2121 - .2640 .9010 .7552 1.3529 25.42 (.2469) (.1652) (.0273) (.1372) (.0259) ( .0538) (.0152) (2 .4676) (2.6874) (. 7407)

9.......... 3 3 5.3696 .9033 -.2009 -.0267 .0439 -.0105 .0182 .0038 .5194 -1.1191 .0763 .7790 1.3116 25.80 (.2533) (.1687) ( .0234) (.0792) (.0226) (.0772) (.0110) (.6184) (l.2498) (. 7084)

10........ ' 4 l 5.5081 .7537 -.1233 .0321 -.0410 .0356 -.0887 .0048 -.9308 1.2946 .3601 .8020 1.0650 25.54 (.1668) (' 1876) (.0234) (.0793) ( .0180) (.0458) (.0150) (1.8119) (1.6637) (.0430)

11......... 4 2 18.3051 1.4369 .0725 -.0763 -.2197 -.0284 .1448 -.0138 -2.2030 .1270 .7757 .7575 1.3464 25.42 (.2515) (.1647) (.0267) (.1377) ( .0256) (.0528) (.0151) (2.4558) (2.6708) (.7408)

12 ......... 4 3 5.8491 .6214 -.1875 -.0267 .1103 .0110 -.0205 .0077 -.3166 -1.2486 .2793 .6795 1.5795 25.80 (.2469) (.1973) (.0348) (, 1318) (.0272) (.0454) (.0242) (2 .6940) (2.9959) (1.2102)

13 ......... 5 1 11.8464 .5816 .0433 -.0225 .1249 -.0038 .0129 -.0054 2.4346 -1.0040 .6820 '7129 1.2826 25.54 (.2564) (.1758) (.0322) (.0957) {.0157) (.0461) (.0100) (2.0212) (l.7834) (1.0087)

14.. ....... 5 2 14. 7115 1.4145 - .0592 -.0585 -.1690 -.0292 .1243 -.0136 -2.0167 .3891 .5451 .7504 1.3789 25.37 (.2640) (.1691) (.0274) (.1409) (.0266) (.0519) (.0156) (2.5760) (2 '7994) (.7733)

15 ......... 5 3 7 .6951 .6665 - .1902 -.0245 .0962 .0101 .0144 .0048 -.5094 -.9439 .2642 .6910 1.5509 25.80 (.2416) (.1930) (.0342) (.1300) (.0267) (.0449) (.0234) (2 .5859) (2.9144) (l.1677)

16 ......... 6 I 11.8860 .6012 .0318 -.0245 .1253 -.0037 .0130 -.0054 2.5070 -1.0917 -.7287 .7237 1.2582 25.54 (.2375) (.1724) (.0312) (.0913) (.0154) (.0448) (.0096) (1. 9188) (I. 7429) (. 9704)

17 ......... 6 2 - 1.2277 1.1974 -.1460 - .0035 -.1514 .0125 -.0021 -.0067 -1.4458 1.6460 .0545 .6639 1.6000 25.37 (.3702) (.2123) (.0334) (.1729) (.0264) (.0622) (.0157) (I. 9524) (2.3319) (.6917)

• For a. one-month projection actual lagged price is used. f The number in parentheses is the standard error of the coefficient.

TABLE 4 (Continued)

PRICE FORECASTING EQUAJl'IONS FOR CHOICE GRADE, 900-1,100 POUND SLAUGHTER STEERS, EL CENTRAO, CALIFORNIA

Constant Lagged prices Projected marketings of fed cattle Quarterly dummy variables*

Equation Lengthof price

projection

Month of

quarter

term PCt-1> Pc•-12> California Arizona Texas Colorado North

central Qi Q• Q, R• Sf/ y

a b1 "' bs ht b, b, In b, b. bio

months dollars/cwt

18......... 6 3 12.4075 .7185 -.1866 -.0178 .0000 .0137 -.0136 -.0010 -.9467 - .0768 .2178 .6965 1.5634 25.78 (.2633) (.1945) (.0352) (.1358) (.0267) (.0454) (.0233) (2.5400) (2.9004) (l.1805)

19......... 7 1 11.1214 .6434 .0076 -.0281 .1203 -.0032 .0094 -.0040 2.3738 -1.3264 - .6967 .7364 1.2518 25.53 (.2438) (.1720) (.0306) (.0903) (.0154) (.0452) {.0098) (1.9174) (1. 7402) (.9734)

20......... 7 2 .6108 1.1549 -.1503 -.0031 -.1257 .0129 .0016 .0046 -1.1495 1.3704 .0186 .6995 1.5868 25.37 {.3468) (.2104) (.0331) (.1671) (.0260) (.0616) (.0154) (1.9073) (2.2856) (.6879)

lll......... 7 3 21.9801 .5564 -.1904 -.0024 -.0138 .0278 .0648 -.0127 - .96B3 -1.7788 - .1159 .6557 1.6653 25.78 {.3493) (.2231) (.0532) {.1792) (.0254) (.0606) (.0086) (l.3834) (2.2317) (1.5087)

22......... 8 1 1- 9.6141 1.1474 .1396 -.0045 .0429 -.0296 .0395 .0001 1.5791 - .3272 .6156 .7069 1.3198 25.53 (.2877) (.1832) (.0280) (.9053) (.0244) (.0478) (.0035) (1.2263) (l.4530) (.9252)

23 ......... 8 2 .4934 1.2034 -.1510 -.0008 -.1519 .0122 .0054 .0040 -1.2143 1.5881 .0317 .6780 1.5951 25.38 (.3604) (.2120) (.0334) (.1758) (.0261) (.0625) (.0156) (1.9843) (2.3638) (.7070)

24......... 8 3 22.1637 .5452 -.1926 -.0024 - .0114 .0284 .0630 -.0125 -.9845 -1.7488 .0995 .6559 1.6648 25.78 (.34io) (.2226) (.0531) (.1785) (.0252) (.0605) (.0086) (1.3762) (2.2343) (1.5024)

25 ......... 9 1 1- 9.2611 1.1371 .1356 -.0079 .0525 -.0308 .0418 .0001 1.7292 - .5222 .7025 .7034 1.3278 25.53 (.2897) (.1842) (.0280) (.0965) (.0248) (.0483) (.0036) (l.2461) (1.4602) (.9390)

26......... 9 2 16.1307 .9483 -.2233 -.0278 -.1300 .0102 .0601 -.0044 - .8380 .0403 1.1911 .6756 1.6009 25.38 (.3338) (.1841) (.0365) (.1127) (.0262) (.0559) (.0054) (.9488) (2.0710) (1.1776)

27......... 9 3 23.1779 .5360 -.2115 -.0041 -.0100 .0293 .0653 .0129 -1.0316 -1.8461 .0667 .6562 1.6859 25.82 (.3289) (.2224) (.0539) (.1796) (.0252) (.0614) (.0087) (1.3590) (2.2489) (1.4962)

28......... 10 1 1- 7.8113 1.1280 .1153 -.0193 .0525 -.0336 .0508 .0006 2.0700 - .9577 -1.0127 .6997 1.3350 25.60 (.2854) (.1866) (.0302) (.0986) (.0252) (.0496) (.0036) (1.2893) (1.5076) (.9959)

29........ 10 2 17 .2424 .9220 -.2316 -.0306 -.1274 .0112 .0597 .0043 - .8282 - .0538 1.2322 .6684 1.6187 25.38 (.3407) (.1869) (.0367) (.1139) (.0266) (.0566) (.0055) (.9600) (2.0927) (1.1894)

30 ......... 10 3 12.7202 .8612 -.2099 - .0594 .0668 .0102 - .0052 .0018 -1.1969 -1.1185 .9638 .6446 1. 7140 25.82 (.3099) (.2317) (.0689) (.1109) (.0221) (.0501) (.0052) (1.0053) (1.6412) (1.4540)

31. ........ 11 2 17 .6374 .8937 - .2315 -.0308 -.1186 .0127 .0546 -.0041 - .7956 - .0827 1.2148 .6578 1.6571 25.44 (. 3446) (.1915) (.0383) (.1172) (.0272) {.0578) (.0057) (. 9892) (2.1424) (1.2248)

32......... 11 3 13.2085 .8658 -.2160 -.0634 .0654 .0103 -.0044 .0020 -1.2342 -1.1866 1.0355 .6412 1.7223 25.82 (.3184) (.2328) (.0691) (.1114) (.0223) (.0503) (.0052) (1.0110) (1.6513) (1.4612)

33 ......... 12 3 12.3549 .8495 -.2062 -.0572 .0713 .0099 -.0043 .0017 -1.1832 -1.0827 .9667 .6381 1.7607 25.85 (.3292) (.2401) (.0738) (.1152) (.0230) (.0515) (.0054) (1.0499) (1.7367) (1.5216)

•For a one-month projection aetual lagged price is used. t The number in parentheses is the standard en-or of the coefficient.

15 Giannini Foundation Monograph • Number £8 • March, 197£

TABLE 5 EQUATIONS FOR ESTIMATING PRICES OF 700- TO 900- AND

1,100- TO 1,300-POUND STEERS GIVEN AN ESTIMATED PRICE FOR 900- TO 1,100-POUND STEERS

Weight group Grade Estimating equation R, Sy

pounda

700 to 900 ............ Choice c

P1_. = c

1.172 + . 960 Po_u .994 .108

1,100 to 1,300 ............ Choice c (]

Pu..... = -.742 + 1.019 p.._11 .985 .202

700 to 900 ...... ..... Good G p,_. G

1. 206 + . 959 p._11 .985 .154

900 to 1,100 ............ Good G

p,_11 = c

.848 + .922 p._,, .988 .226

1,100 to 1,300.... """ .. Good G G

Pu...1• = - .467 + 1.007 Po_n .985 .200

Other weights and grades of slaughter cattle

To analyze the decision problems, price estimates are also required for other weights and grades of slaughter cattle as well as for feeder cattle. Analy­sis of historical data indicates that prices for other weights and grades of slaughter cattle can be derived from the prices for 900-to 1,100-pound Choice grade slaugh­ter steers. The estimated relationships between these various prices are given in table 5. Thus, once the price is pre­dicted for the 900- to 1,100-pound Choice animals, the required prices of other classes can be estimated.

Feeder cattle

Feeder cattle prices were predicted by relating prices of 550-to 750-pound Good grade feeder steers at Stockton to their lagged values and to predicted slaughter cattle prices. These equations are given in table 6.

The price forecasting model for slaugh­ter steers with the accompanying re­lations between slaughter cattle prices for other weights and grades provides the basic foundation for marketing the decision problems studied under Models I and II. The feeder cattle price fore­casting equations are added in Model III to give the format for purchasing de­cisions.

PROBABILITY DISTRIBUTIONS

The marketing decision rules de­veloped in this study are based on ex­pected price changes rather than ex­pected price levels, although the latter also affect the . marketing decision. As will be shown later, each price level is considered as a separate decision prob­lem. Therefore, at this point, our con­cern is with gaining information about price changes that may occur.

In the following s.nalysis, two market­ing strategies are developed based on probability distributions of historical

price data. The first strategy simply uses t}fe marginal probability distribu­tion of historical prices. This distribu­tion is referred to as the "a priori" dis­tribution following the terminology of the no data models outlined in the pre­vious section on the decision model.

The second distribution is the condi­tional probability of a price change given the magnitude of the price change predicted by the price forecasting model. This distribution is hereafter referred to as the "a posteriori" distribution which

TABLE 6 PRICE FORECASTING E~UATIONS FOR GOOD GRADE, 550-750 POUND FEEDER STEERS, STOCKTON, CALIFORNIA

Length of price

projection Month of quarter

Constant term

a

Quarterly dummy variables

PP(l_I) f>,

l 2nd 1,~arter Jbi b2 1st quarter 3rd quarter ba b,

months

1. ............... 1 1.0770 .7208 .2176 .1501 .1592 - .2946 (.1320) (.1282) (.1696) (. 1762) (.1688)

1. ............. 2 .4930 .5910 .3621 .2033 - .1841 -.4741 (.1507) (.1172) (.2040) (.2172) (.2024)

1................ 3 -1.0475 .7978 .2306 .2733 -.4469 -.1281 (.0954) (.0783) (.1313) (.1289) (.1276)

2................ 1 1.4486 .5545 .3595 .1010 .2510 -.3456 (.1629) (.1639) (.2321) (.2460) (.2284)

2................ 2 2.0339 .2671 .6061 .1623 .0074 -.5835 (.1929) (.1498) (.2752) (.3021) (.2726)

2................ 3 - .1371 .4866 .4841 .2553 -.3320 - .1787 (.2175) (.1726) (.2310) (.2356) (.2245)

3. ''.'.'.'.'''''. 1 1. 7004 .3661 .5245 -.0348 .3929 -.4032 (.1768) (.1840) (.2532) (.2696) (.2681)

3 ................ 2 2.2279 .1181 .7382 .1398 .0924 -.6466 (.2416) (.1926) (.3051) (.3398) (.3050)

3 ................ 3 l.0267 .1923 .7116 .3445 -.3502 -.2181 (.3927) (.3038) (.2763) (.2805) (.2685)

4................ 1 1.4276 .3690 .5327 - .0624 .4500 -.4383 (.1655) (.1767) (.2472) (.2666) (.2696)

4 .... , ........... 2 1.9080 .2061 .6684 .1519 .0427 -.6140 (.2609) (.2012) (.3138) (.3523) (.3240)

4................ 3 .4558 .1465 .7768 .3408 .3384 -.2192

A ............... 1 .3739 (.4037) (.3225) (.3054) (.3116) (.2969) .3509 .5919 -.0308 .4329 -.4737

(.1702) (.1859) (.2725) (.2895) (.2880) 5 ................ 2 1.6250 .2732 .6169 .1585 .0606 - .6398

( .2902) (.2266) (.3176) (.3701) (.3362) 5 .. '.'.'.'.' ..... 3 l.1114 .0648 .8280 .3360 .3325 -.1894

( .4545) (.3593) (.3109) (.3168) (.3159) 6........ ' ....... 1 .1489 .3751 .5780 - .0105 .4073 -.4729

(.1674) (.1792) (.2734) (.2899) (.2844) 6................. 2 .3891 .3890 .5584 .1547 -.0204 -.5701

(.4480) (.3562) (.3171) (.4221) (.3446) 6................ 3 1.8227 -.0621 .9181 .3507 -.2860 -.2386

(.5117) (.4012) (.3162) (.3251) (.3257) <'cc,

Shift variable

uni1

bo

-1.2268 (.5411)

-1.5767 (.7014)

.1661 (.4540)

-2.0771 (.6960)

-2.9721 (. 9379)

-1.6386 (1.0358)

-2.8240 (. 7620)

-3.5278 (1.1267)

-3.0114 (1.8341)

-2.8438 (.7071)

-3.2420 (1.2281)

-3.3018 (1.9324) -2.9296

(.7244) -3.0046

(1.3445) -3.7607

(2' 1724) -2.8134

(.7217) -2.7187

(2,0844) -4.3962

(2.4508)

R'

.9601

.9422

.9754

.9276

.8982

.9269

.9125

.8751

.8982

.9153

.8737

.8746

.8999

.8710

.8768

.9039

.8772

.8733

Sy

1,000 head

.5665

.7094

.4588

.7692

.9531

.8033

.8593

l.0555

.9477

.8455

1.0789

1.0518

.9188

1.0900

1.0585

.9145

1.0635

1.0731

17 Gianninii Foundation Monograph • N~imber 28 • March, 1972

TABLE 7 POSSIBLE PRICE CHANGES AND

vALUES OF e USED IN THE STUDY

Interval Value of 0 State

dollars per cwt

0 ,..... .. . ... 1.26 :$\ aP 1.50

i. 25e.. .. . . . .. . . .. .76 :$\ aP"" i.oo0.. .. . . . ...... .26,,;; /J.P"' .75 .50 0 ,..... . . . . . - .2s s aP ;;,; .2s .oo

26 50e............. .15 ,,;, ap ;;,; · - · 9 25

•···· · · · · · · · · -i. s !~; _1::: =~:~81 ........... .

provides the basis for the data problem. Both probability distributions are de­

rived from the same set of price data. To this extent therefore, one could look ' . upon the a priori distribution as a naive price forecasting tool, and the a posteriori distribution as being obtained from a more sophisticated predicting mecha­nism. However, rather than adopting a new terminology at this point, the nota­tion a priori and a posteriori will be retained in following with the original model formulation.

Average monthly price data for the 1960 to 1969 period (i.e., 120 price ob­servations) are used to calculate the probabilities. Seven ranges of price changes (b.P Pct+1l - Pt) 11 are con­sidered in the study shown in table 7. Thus there are seven points in the sam­

' pie space over which the probability dis­tributions are to be calculated. Since we are dealing with a discrete set of possible price changes, the probability distribu­tions can be expressed as a table of values. These values (P,) indicate the probability of a particular value of 0 (price change) occurring and have the following properties (a) 0 S P, S 1 for all i, and (b) 2: P, = 1.

The likeliho~d of a given price change probably is not independent of the cur­rent level of prices relative to some

11 t now refers to the current month.

"normal" or "average" level of prices. If current prices are high, a large drop in prices is more likely than if current prices were well below average. There­fore, it would be desirable to estimate conditional probability distributions for each price level. However, given data limitations, there are not enough obser­1 l te vations in each group to ca cu a mean­ingful probabilities for each of the values of 0. Therefore, the same distribution

of price changes is used to represent all price levels. ·

A priori distribution

The a priori probability distribution presents the information about the ?~e­lihood of 0 available to the dec1s1on maker without any experimentation. It can be derived (either objectively or subjectively) from a historical distri­bution of 0.

In this study, the a priori distribution is defined as the relative frequency of 0 over the 1960 to 1969 period. Calcula­tion of the a priori distribution consists of two steps: (a) calculate 0 = b.P Pct+il - Pt 12 for all available data and (b) determine the relative frequency with which each value of 0 occurred. For example, table 8 shows that for a one-month price change, 0 = 0 (i.e., - .25 < b.P < .25) in 26 out of the 120 month; obs;rved and 0 = $1.50 13 times. These frequencies are used to construct the probability distributions presented in table 9. On the basis of the distribution of 0 over the 1960 to 1969 period for one-month price changes, we would expect 04 (i.e., 0 O) to occur about 22 per cent of the time, 01 (i.e., 0 = $1.50) to occur about 11 per cent of the time, and so on.

Table 9 summarizes, in objective form, the information about the distri­

. . _ _ p ) 12 This refers to a one-month price change. For a two-month price change, l:l.P - (P<t+z) i •

18 B1-0lloolc and Logan: Cattle Feedlot Marketing Deoisions Under Unoertainty

TABLE 8 FREQUENCY OF OCCURRENCE OF PRICE CHANGES OVER THE 1960-1969 PERIOD

Price change

Period covered 01

1.50 02

1.00 0.

0.50 0.

0.00 0,

-0.50 e,

-1.00 07

-1.50

One-month ......... 13 14 21 26 22 12 12 Two-month ......... 26 10 21 15 9 11 28

bution of e available from the historical record of price changes. With the ex­ception of 1968 and 1969, prices of 900­to 1,100-pound slaughter cattle at El Centro over the period studied fluctu­ated around an average level of about $25.00. The lack of trend is shown by the expected value of e, given the a priori probability distribution. The a priori expected price change, E(®) :E !

®; P(®;), is .025 for a one-month

price change, and .017 for a two-month price change. Thus, the a priori informa­tion states that lacking any other in­formation, the cattle feeder reasonably can base his decisions on a zero or very small expected price change.

A posteriori distribution

The a posteriori distribution repre­sents the added information from the price forecasting model available to the decision maker at the time the decision is made. Given a predicted price, P<H-lh the experiment consists of calcu­lliting Z = P(t+ll - P1. Z functions as an estimator of 0 for the coming month(s).13 The accuracy of Z as a pre­

dictor of 0 depends on the accuracy of the price forf3casting model.

The a pos,teriori probability distribu­tion P(®IZ) yields the probability that a. particular value of e (price change) will occur, given Z as a prediction of e. As explained earlier, the conditional probability of P(®IZ) can be calculated by Bayes' formula when the a priori distribution is developed separately from the a posteriori distribution. If the data set for both distributions is identical, as is the case here, then the conditional distribution of e1z can be calculated ;, directly without the use of Bayes' for- · mula. As long as the observations are the same, then the direct approach and the Bayesian formulation will yield identical answers.

Bayes' formula, however, permits the use of additional information which may be available for the estimation of one probability distribution (say P(®)) but not for the direct estimation of the con­ditional d.istribution. Thus, a feeder could substitute his own a priori distri­bution into the above analysis through use of Bayes' formula should he so de­sire.

TABLE 9

A PRIORI PROBABILITY OF PRICE CHANGES

Period Price change

covered 01 e. e, 0. 0, e. a,

One-month ....•..•• .108 .117 .175 .217 .183 .100 .100 Two-month ......... .217 .083 .175 .125 .075 .092 .233

u For two-month estimates of 0, Z is calculated as (P (1+2l - Pt).

19 Giamnini Foundatwn Monograph • Nwmber 28 • March, 1972 ~· . \

TABLE 10

FREQUENCY WITH WHICH Zk WAS PREDICTED GIVEN THAT 0; WAS THE TRUE STATE OF NATURE

Predicted price change

Actual price change z, z, z, z, z, z, Z1

!':50 LOO 0.50 0.00 -0.50 -L50 -1.SO Total

01 1.50........... 6 2 1 13 02= 1.00........... 4 6 3 14 e, 0.50 ........... 5 5 3 20 0. = 0.00 ........... 7 8 8 26 e, = -o.50 ........... 2 9 9 22 0, = -i.oo ........... 3 4 4 12 e, = -i.5o. 2 3 4 12

Total. ................ 6 16 22 31 27 13 119

P(E>[Z) is obtai:rv~d by evaluating the performance of z ~s a predictor of e over the historicaJ p~riod. That is, using the predicted vai~f)s generated by the forecasting model and actual historical data, we can constri,ict a table showing the frequency wit4 ·which actual price changes, @, too!{ pn certain values, given alternative predicted values, Z. This information ii:i presented in table IO for one-month prip~ changes over the 1960 through 196~ period. For example, during this period Z4( -25 :;; l::i.P :;; .25) occurred 31 timeEJ; the predicted inter­vals were correct ei~ht times, too high 14 times and too low nine times. These frequencies are u'.s~d to determine the

conditional probabilities shown in table 11.

The a posteriori distribution is also needed for two-month price changes. The procedure used to calculate this distribution is the same as that for the one-month price change with one modi­fication; the price intervals used to esti­mate the probabilities are increased to one-dollar magnitudes rather than 50 cents. This change results from the in­creased frequency of larger price changes as the time period increases.

Therefore, the number of possible price changes (defined as</> for Model II) are as follows:

. TABLE 11

A P()STERIORI DISTRIBUTION P(0jZ) = P(ZJ~~(Pe)

OF ONE MONTH PRICE CHANGES

Actual pricechange

Projected price change

Z1 L'OO

z. 1.00

z, 0.50

z. 0.00

z, -0.50

z, -1.50

Eh= 1.50........ . .666 .375 .091 .032 02 1.00 ........ . .107 .250 .273 .097 e, = o.5o....... . .i67 .375 .227 .161 .111 e. = o.oo ........ . .318 .258 .296 .230 0s·= -0.50 ........ . .091 .290 .334 .154 e, = -i.oo ........ . .097 .148 .308 .250 Eh -L50 ........ . .065 .111 .308 .750 P(Z) . .............. . .050 .136 .185 .261 .226 .108 .034

20 Bulloc"k and Logan: Cattle Feedlot Mar"keting Decisions Under Uncertainty

Value Interval of cf>;

c/>1 $ .51 :=:; LlP :=:; $ 1.50 $ 1.00 c/>2 $- .50 :=:; LlP :=:; $ .50 0 cf>3 $-1.50:=s;<lP:=s;$- .51' -1.00

The two extreme intervals (c/>1 and cf>3) are actually open-ended in the sense that they include price changes of mag­nitudes greater in absolute terms than $1.50. The use of a representative value for cf>; generally does not affect the analysis; however, some degree of cau­tion is needed in making decisions in­corporating the price forecasting model when the price forecast is for a price

TABLE 12

PROBABILITY DISTRIBUTIONS FOR TWO-MONTH PRICE CHANGES

A posteriori distribution

Actual pricechange

P(<t>IZ)

A priori distribution

P(<t>) Predicted price change

z, Z2 z, 1.00 0.00 -1.00

<1>1= 1.00.... .814 .296 .021 .383 <1>2= 0.00.... .186 .333 .250 .242 <l>s = -1.00.... .000 .371 .729 .375

decrease of more than $1.50. This factor will be discussed later. The a priori and a posteriori distributions for Model II are presented in table 12.

FEEDLOT PERFORMANCE OF CATTLE

Two additional sets of information are required for the decision models: (a) the cost per pound of gain and (b) the expected carcass grade at alternative slaughter weights. In the general formu­lation of the decision problems, these two factors could be incorporated as additional states of nature, W and U. However, most experienced cattle feed­ers can estimate accurately the values of these variables. In this analysis, we assume that the errors associated with the feeder's estimates are distributed about the true value with a very small variance and that the errors, therefore, a,e negligible. These relationships have been incorporated into the decision mod­els at their average (expected) value. To the degree that these assumptions are not met, a greater needed expected price increase than suggested by the following model would be required. The situation will be discussed later in the section on sensitivity of the model.

Estimation of cost per pound of gain

Two sets of cost estimates are needed: (a) an estimate of the average cost per

pound of gain over the normal feeding period (defined here as feeding 600­pound feeder steers to 1,000-pound slaughter weight) in order to calculate break-even feeding margins for the pur­chasing decisions and (b) break-down of cost per pound of gain as the weight of the animal progresses from 600 to 1,200 pounds-for example, an estimate of the cost per pound of gain for feeding 950-pound steers another 30 days vs the cost per pound of gain for feeding 1,100-pound steers another 30 days. Since the cost per pound of gain in­creases with the weight of the animal, specific information about this rela­tionship is needed to make marketing decisions for alternative weights of cat­tle. The information is needed for weights up to 1,200 pounds, since marketing de­cisions generally are made for slaughter cattle ranging up to this weight.

The derivation of these estimates con­sists of (a) specifying a typical finishing ration and determining its net energy content, (b) estimating daily feed con­umption for various weight groups, (c)c salculating the resulting gain per day for

21 Giammli1niFoundatwnMonograph • Numbe'r 28 • March, 1972

each weight group using the tables in Lofgreen and Garrett (1968), and (d) calculating the feed cost per pound of gain, using average feed prices. Nonfeed costs are then added to feed costs to ar­rive at total cost per pound of gain.

Ration. The Hopkin and Kramer (1965) survey indicates the average fin­ishing ration for cattle feedlots in Cali­fornia has the following composition: roughages, 15.3 per cent; feed grains, 57.6 per cent; other high-energy feeds, 14.5 per cent; protein supplement, 3.7 per cent; molasses, 5.7 per cent; fat, 2.1 per cent; and minerals, 1.1 per cent. Alfalfa, barley and milo, beet pulp, and cottonseed meal were the most widely used feeds in their respective categories. Using this information, the following ration was formulated .as the "typical" finishing ration.14

Percentage of Feed total ration

Alfalfa 15.3 Barley 42.3 Milo 15.3 Beet pulp 14.5 Cottonseed meal 3.7 Molasses 5.7 Fat 2.1 Minerals 1.1

TOTAL 100.0

This ration provides 79.95 megcal. of net energy for maintenance (NE,,.) and 50.69 megcal. of net energy for produc­tion (NEP) per 100 pounds of feed. 15 It also provides all of the nutrient require­ments, except vitamin A for fattening

yearling cattle, as specified by Morrison (1952). The latter requirement can eas­ily be met by adding a vitamin supple­ment to the ration. The Morrison stand­ards and the nutrient levels supplied by this ration are shown in tables 13 and 14.

Feed consu:mption. The National Academy of Sciences, NRC (1963) re­port on nutrient requirements for beef cattle indicates that.yearling cattle on.a finishing ration consume from 17.5 pounds of feed per day at 600 pounds body weight to 25.8 pounds at 1,100 pounds. These intake levels range from 2.9 percent to 2.3 percent of body weight. The report further states, "Fin­ishing cattle consume feeds in amounts equal to 2.2 to 3.0 per cent of their live weight, dependent upon the concen­trate-roughage ratio and age and con­dition of animals. Older cattle and more fleshy individuals consume less feed per unit of body weight than do younger animals carrying less condition ... As concentrate content increases, feed con­sumption is usually reduced."

The following "rule of thumb" for determining feed consumption appears to be consistent with the NRC state­ment.

Daily feed Percentage of consumption as concentrate percentage of in ration live weight

90 2.4 to 2.6 80 ~o 90 2.6 to 2.7 70 to 80 2.7 to 2.8

<70 2.8 All roughage 2.9

14 The ration used here can be varied to achieve different rates of gain or to take advantage of changes in price relationships among the various feed components. Since many feedlots carry sizable storage of grain and hay (King, 1962), however, the specification of a particular ration for a period of one month does not appreciably affect the nature of the problem solution.

10 The net energy characteristics of a ration are determined by the amount of each feed used in the ration. Each feed has a given net energy content for maintenance and for gain. The net energy figures used in these calculations were obtained from Lofgreen and Garrett (1968, pp. 24-25).

22 Bullock and Logan: Cattle Feedlot Marketing Deaiswns Under Uncertainty

TABLE 13 MORRISON FEEDING STANDARDS: PER HEAD DAILY

REQUIREMENT FOR FATTENING YEARLING

Weight of animal

600.. .......... 700.............

800. ·········· .. 900 .............

1,000............. 1,100......... ' .. '

Dry matter

15.0-17.6 16.5-19.l 17 .8-20.4 18.9-21.7 20.0-23.0 21.0-24.0

Digestibleprotein

pounds

1.18-1.32 1.36-1.52 1.52-1.68 l.64-1.82 1.71-1.91 1.76--1.96

Total digestiblenutrients

10. 7-12.3 12. 7-14.3 14.1-15.9 15.4-17.2 16.0-18.0 16.5-18.5

Calcium

.044

.044

.044

.044

.044

.044

Phosphorus Carotene

mg

.037

.040

.042

.044

.044

.044

35 40 45 50 55 60

TABLE 14

NUTRIENTS SUPPLIED BY "TYPICAL" RATION: BASED ON DAILY FEED INTAKE SHOWN IN TABLE 15

Weight of animal

Dry matter

Digestibleprotein

Total digestiblenutrients

Calcium Phosphorus Carotene

600 ....... ' .. 700 .... ' ... ' .. 800 ............ 900.. ' ... ' .. ' ..

1,000............ 1,100...........

14.53 16.35 17.99 19.46 20.76 21.88

pounds

1.34 11.67 1.51 13.13 1.66 14.45 1.80 15.63 1.92 lll.67 2.02 17.57

.064

.072

.080

.086

.092

.097

.046

.052

.057

.062

.066

.070

mg.

23.06 25.94 28.55 30.89 32.94 34.73

These two sources of information are combined into three constraints for estimating daily feed consumption: (a) daily consumption rates vary from 2.2 to 3.0 per cent of body weight; (b) daily feed consumption as percentage of body weight declines as liveweight increases; and (c) average consumption rate over the normal feeding period (600 to 1,000 ~unds) should average about 2.6 per cent of body weight, given the above 85 per cent concentrate ration.

In this study, a consumption rate of 2.6 per cent was assigned to 800-pound animals (the midpoint for the 600- to 1,000-pound normal feeding interval); then linear interpolation consistent with constraints 1 and 2 was used to estimate consumption rates for the remaining weight groups. The consumption rates are then applied to the liveweight to arrive at estimates of daily feed con­

sumption shown in column 3 of table 15· These consumption estimates are

slightly below those presented in the NRC report. However, since the ration used here is a higher energy ration than the NRC ration, the results appear to be consistent.

Daily gain. The daily intake of net energy and the resulting gain per day were calculated using the tables in Lof­green and Garrett (1968). The average daily gain from 600 pounds to 1,000 pounds is 2.77 pounds. This level again is slightly higher than those reported in the NRC report; however, it seems reasonable, given the higher energy con­tent of the ration being considered.

The average daily gains shown in table 15 compare favorably with the re­sults of a survey reported by Logan and King (1966) in which. animals placed on feed weighing 600 to 799 pounds had an

TABLE 15

RESULTS OF FEEDING "TYPICAL" RATION, ASSUMING A NORMAL FEEDING PERIOD OF 600 TO 1,000 POUNDS

Weight of feeder steer

Daily feed consumption as percentage

of body weight

Feed consumption

per day

Daily maintenance requirements

NE,.

Feed per dayrequired for

maintenance*

Feed available r,erdayor gain

NE,tavailable per day

Dailygain

Feed per poundof gain

Feed cost per pound of gain

pounds per cent pounds megcal pcunds pounds megcal pound• pounds cents

600 2.80 16.80 5.21 6.52 10.28 5.21 2.80 6.00 15.53 650 2.75 17.88 5.53 6.92 10.96 5.56 2.81 6.36 16.46 700 2.70 18.90 5.85 7.32 11.58 5.87 2.80 6.75 17.47 750 2.65 19.88 6.16 7. 71 12.17 6.17 2.80 7.10 18.37 800 2.60 20.80 6.47 8.10 12.70 6.44 2.79 7.46 19.31 850 2.55 21.68 6.77 8.47 13.21 6.70 2.77 7.83 20.26 900 2.50 22.50 7.06 8.83 13 .67 6.93 2.75 8.18 21.17 950 2.45 23.28 7.36 9.21 14.07 7 .13 2.72 8.56 22.15

1,000 2.40 24.00 7.65 9.57 14.43 7 .31 2.69 8.92 23.08 1,050 2.35 24.68 7.93 9.92 14.76 7.48 2.65 9.31 24.09 1,100 2.30 25.30 8.21 10.27 15.03 7.62 2.61 9.69 25.08 1, 150 2.25 25.88 8.49t 10.62 15.26 7.74 2.57 10.07 26.06 1,200 2.20 26.40 8.m 10.97 15.43 7.82 2.52 10.48 27.12

•NE.,+ .7994. t (.5069) (feed available for gain). ~The Net Energy requirement for steers over 1, 100 pounds is not presented in Lofgreen and Garrett (1968). The values shown here were supplied by G. P. Lofgreen.

24 Bulloc."k; amil Logan: Cattle Feedlot Marketing Decisi-Ons Under Uncertainty

average daily gain of 2.68 pounds per component of purchasing and marketing day over the feeding period. If the aver- decisions. The expected slaughter grade age daily gain from table 15 is extended clearly affects the expected feeding to cover weights above 1,000 pounds, the margin and, therefore, the price the cat­estimated gains correspond even closer · tle feeder can afford to pay for feeder with the survey results. cattle. Similarly, the extent of upgrad­

Cost per pound of gain. The typical ing achieved by feeding another 30 days ration cost $2.588 per 100 pounds at directly affects expected returns from average feed prices for 1967, yielding a extending the feeding period. feed cost per pound of gain from 15.33 Carcass grade is related to the con­cents per pound for 600-pound steers to formation of the animal and the degree 27.12 cents per pound for 1,200-pound of finish. Some animals with good con­steers. Feed cost per pound of gain formation may grade Choice at 700 averages 19.31 cents per pound over the pounds or perhaps less; other animals 600- to 1,000-pound feeding range. may not attain Choice grade even if fed

King (1962, p. 30) has estimated non­ to 1,200 pounds. Conversely, conforma­feed costs of 5.46 cents per head per day tion alone is not sufficient for grading for feedlots with capacity of 11,280 head Choice. operating at 80 per cent capacity.16 If Given steers of a specified age and cattle gain 2.77 pounds per day, the conformation, the expected carcass grade normal feeding period (600 to 1,000 is a function of the weight at which the pounds) would be 144 days, giving a animals are slaughtered. Slaughter nonfeed cost per pound of gain over the weight serves as an indicator of finish normal feeding period of 2.31 cents. for steers of similar conformation. Finish Thus, the estimated average total cost of refers to a combination of fat covering gain over the 600- to 1,000-pound feed­ and marbling (i.e., presence of fat in the ing interval is 21.62 cents per pound. muscling) and is not a linear function of This estimate of total cost per pound of weight. The proportion of additional gain is used to calculate break-even weight added as fat covering (relative margins for the purchasing decisions, to additional marbling) increases as the because. we assume that these decisions weight of the. animal increases. Beyond are based on expected returns from a a minimal standard added fat covering normal feeding period. To the extent does little to improve the grade of the that cattle are fed to weights above carcass. Therefore, we would expect the 1,000 pounds, this estimate will be bi­ probability of attaining Choice grade to alJed downward. increase at a decreasing rate as the

weight of the animal increases. Carcass grade-Slaughter '\yeight At any given weight of an animal, therelationship probability of its attaining a certain

Prices of Choice grade slaughter steers grade can be considered as discrete, that are generally above Good grade prices. is, at each possible weight the steer will Therefore, the carcass grade that feeder either grade less than Choice or will steers can be expected to attain at the grade Choice or better. On the other end of the feeding period is an important hand, the relationship between grade

16 A capacity of 11,280 head is assumed to be a representative size feedlot. Cost figures for 80 per cent capacity are used because published data indicate only about 75 per cent of the total feedlot capacity in California is utilized. The nonfeed costs include management and office, taxes, insurance, interest, depreciation, death loss, labor, and utilities.

25 Giannini Foundation Monograph • Number 28 • March, 1972

and weight can be expressed as a con­tinuous function. By expressing the per­centage grading Choice or better as a function of average weight of the lot, a continuous relationship between carcass grade and slaughter weight can be de­rived. Because it is a continuous func­tion, we have a measure of the degree of upgrading that can be expected from feeding cattle from one weight to an­other as well as the percentage of ani­mals that can be expected to grade Choice or better at alternative weights.

The weight-grade relationship is based on 2,080 observations of slaughter weight and carcass grades of cattle. The data consist of 487 observations from cattle feeding experiments performed by the University of California and 1,593 ob­servations from two northern California slaughter plants.17 The data obtained from the slaughter plants were in terms of carcass weight and grade. Carcass weights were converted to live weights by using the following dressing per­entages:

Dressing Carcass weight percentage

< 550 pounds 59.5 551 - 600 pounds 60.0

> 600 pounds 60.5

For example, a 580-pound carcass weight is converted to live weight as follows: (580 + .60) = 967 pounds. The slaugh­ter weights were grouped into 50-pound intervals with the midpoint for the in­terval ranging from 725 pounds to 1,275 pounds. The percentage of cattle grad­ing Choice or better was then calculated for each weight interval and plotted against the average weight of the inter­

val (figure 1). If the resulting live weight calculated from carcass data was With ±2 pounds of the interval boundaries, the observation was discarded to reduce errors of placing a carcass in the wrong interval.

The sample data represented by dots in figure 1 show that the percentage grading Choice increases rather rapidly from 725 pounds to about 1,000 pounds and levels off beyond 1,000 pounds. The data also indicate that the function is asymptotic at some value less than 100. That is, some of the cattle fed will never attain Choice grade regardless of weight.

Two simple equational forms were fitted to the data: (a) Y =a+ bVX, and (b) Y =a+ log.X; where Y is the percentage grading Choice or better and X is the average weight of cattle in each weight interval. In both cases the re­sulting equations are essentially linear over the range of data because of the rather large magnitude and relatively narrow range of the values of X (weights). However, this nonlinear re­lationship in the square root and semi­log functions is most prevalent when the values of X are small and there is a relatively wide range of X values con­sidered. For large numbers, VX and log. X approach linearity, particularly over short ranges.

To avoid this linearity, the scale of the X axis''was changed to reduce the absolute magnitude of the X values and thus increase the relative range of the dependent variable. This was accom­plished by forming a new variable Z = -13.5 + .02X. For example, if X = 725, z = -13.5 + .02(725) = 1. If x = 775, Z = -13.5 + .02(775) = 2, and so

17 The use of these data to estimate the relationship between carcass grade and slaughter weight requires two major assumptions regarding the homogeneity of the data both within and among the two sets: (1) that the animals observed were of similar quality when placed on feed and are representative of the quality of cattle fed in California, and (2) that the feedlots in which these cattle were fed are typical of California feedlots in feeding programs followed and rations fed. We have no information to substantiate or repudiate either of these assumptions.

26 Bulwak and Logan: Cattle Feedlot MMketing Decisions Under Uncertainty

80

7(j

& 60.. ..a.. ..0

50 Y=-.00145 + .28845 log, Z .!:! 0..s:: u Cl c 40

"5 e Cl

c 30., u

~ ll.

> 20

10

0 .._~-'-~-'-~--'~~'--~-'-~-'-~--'-~--'-~~'--~"'--~-'-~-'--~ x 725 n5 825 875 925 975 1025 1075 1125 1175 1225 1275 Weight

z l 2 3 4 5 6 7 8 9 10 11 12

Fig. 1. Relationship between slaughter weight and carcass grade.

forth. Y can now be expressed as a function of Z instead of the original variable X.

Both the square root and semi-log functions were fitted to the adjusted data. The semi-log function has the highest degree of explanation and lowest standard error. The resulting equation is as follows:18

Y = .00145 + .28845 log. Z (.21635)*

R2 = .969 (11)

, Bu= .041

where: Y is the percentage grading Choice or better. Z = -13.5 + .02X; Xis the average sl!!-ughter weight of cattle in the lot. * = standard error of coefficient.

This equation is plotted against actual observations in figure 1.

Using equation 11, we can obtain an estimate of the percentage of animals ~ in a particular lot that can be expected to grade Choice or better, given the average weight of the cattle in the lot. For example, if the average weight of a lot of cattle is 1,050 pounds, the ex­pected percentage grading Choice is estimated as follows:

z = -13.5 + .02(1,050) = 7.50

log. Z = 2.0149

y = - .00145 .28845(2.01490) = 58.0 per cent.

The relationship estimated here is consistent with information provided by the Hopkin-Kramer survey (1965) in which 58 per cent of the fed cattle marketed in California in 1963 were Choice grade.

1sThe square root function is: Y = - .19312 + .27575 Vz R2 = .945 Sv = .054; where Y and (.02101)*

Z are defined above. *Indicates standard error of coefficient.

27 Gia111nini Foundation Monograph • Number 28 • March, 1972

DECISION RULES Four models are developed in this

section. As stated before, model I uses Bayesian decision theory to arrive at a set of short-run marketing decision rules based on a one-month prediction of slaughter cattle prices. Model II ex­tends the marketing decision rules to cover two-month price predictions. ·Model III incorporates the information provided by the price forecasting model to develop criteria for purchasing feeder cattle. Model IV combines the market­ing and purchasing decision rules into a six-month planning model.

Feeding decisions for one month­Model I

The decision to feed a particular lot of cattle another 30 days or sell them at current weight is based on expected prof­it from extending the feeding period.19

A particular lot of cattle will be con­tinued on feed only if expected marginal profit from additional feeding is positive.

The outcome, A (marginal profit per head) of extending the feeding period another 30 days is defined by equation (12).

>.. = GV- [CV C]

>.. = P<t+il(W + G) - (PtW + C) (12)

>.. = P<t+ilG +.&PW - C

A= (aP+Pe)G .&PW-a

where:

GV = gross value of steer 30 days hence (dollars per head)

CV = current value of steer (dollars per head)

C = cost of feeding the steer an­other 30 days (dollars per head)

Pt = current price (dollars per pound) P (t+il = price 30 days hence (dollars

per pound) ..&P = (P(l+1l - Pi) = price change

during next 30 days (dollars per pound)

W = current weight of steer (pounds) G = gain from feeding another 30

days (ptmnds)

Thus, the marginal profit from ex­tending the feeding period another 30 days is the value of the weight gained ( aP P 1)G, plus the change in value of the current weight (.&PW), minus feeding cost.

As shown previously, daily gain and feeding cost both are a function of the weight of the steer. Consequently, the outcome of extending the feeding period can be expressed as a function of (a) current weight of the steer, (b) current slaughter cattle price, and (c) the price change that occurs. Current weight and current slaughter cattle prices are known at the time the decision is made to sell or continue feeding. Thus, the uncer­tainty about the outcome of a decision to extend the feeding period arises from the price change that may occur.

The price change, ..&P, has two com­ponents: the change in the level of slaughter cattle prices and an increase in price received resulting from upgrad­ing. The upgrading has the effect of in­

. creasing ..&P since prices of Choice grade slaughter cattle average about one dol­lar per hundredweight above Good grade prices. Thus, in evaluating equa­tion (12), explicit attention must be given to the degree of upgrading that

19 The decision rules are based on expected returns to the "average" steer in a particular lot of cattle. The rules are applicable to marketing and purchasing decisions for the entire lot of cattle of specified average weight rather than to individual animals.

28 Bulloo71i llfltd Logan: Cattle Feedlot Mar71ieting Decisions Under Uncertainty

can be expected by incorporating the weight,;glaughter grade relationship ex­pressed by equation (11).

The Bayesian statement of the, de­cision problem is as follows: Given the average weight of a particular lot of cat­tle and the current price of slaughter cattle, the cattle feeder is faced with two possible actions-Ai =feed the cattle

another 30 days or A2 = sell the cattle at current weight. The outcome (A.i1) of these decisions depends on the change in the price of slaughter cattle (state of nature 0 1-) that occurs in the next 30 days. Using the range of price changes (states of nature) defined in table 14, the decision problem can be formulated as follows:

Actions

States of nature El = [P <1+1J - Pt] (dollars)

Ela 0.50

@5

-0.50 06 1.00

01 1.50

A1 (feed) A2 (sell)

Ai; is the outcome (marginal profit) of following action Ai, given the occur­rence of 01. A.21 = 0 for all values of j, because marginal profits from selling at current weight are zero. A.11 ~ 0, de­pending on the price change thatoccurs. The cost of feeding, C, is the same re­gardless of which value of e that occurs.

The "no data" strategy is derived us­ing the a priori distribution, P(El), and is defined as selecting the action for which fr, = 2; Ai;P(El;) is the greatest,

j

where fr, is the expected profit of the i 1"

action.20 The "data" strategy utilizes the a posteriori distribution, and the optimal action is the one for which '1i'> 2; A.;;P(El1JZ.,) is the maximum,

j

where z., is the predicted price change. Derivation of decision rules. Cur­

rent price and current weight are held constant in the above formulation of the problem; however, these factors directly affect the outcome of the decision (equa­tion 12). Therefore, decision rules must be derived for alternative combinations

of prices and weights. Nine weight categories and eight price levels are con­sidered, as shown in table 16. Thus, 72 separate decision problems are con­sidered, one for each combination of weight and current prices.

The price levels in table 16 are for 900- to 1,100-pound Choice grade slaughter steers. All statements regarding price levels and price changes refer to prices of900- to 1,100-pound Choice grade steers. By using the price relationships presented in table 5 to evaluate the out­come of each action, we can state the re­sulting decision rules in terms of current levels and price changes for 900- to 1,100-pound Choice grade slaughter steers.

The first step in developing the de­cision rules is to evaluate the outcome, Ai;, of each action under alternative states of nature (price changes). The second step is to calculate the expected profit from each action utilizing the a priori and a posteriori probability distri­butions. Since A.2; 0 (i.e., the outcome

20 Throughout this study we are assuming that the cattle feeder's utility function is linear with respect to money over the relevant range. Thus decision rules are developed by maximizing expected profits which is equivalent to maximizing expected utility.

29 Giannini Foundation Monograph • Nwmber 28 • March, 1972

TABLE 16

WEIGHT CATEGORIES AND PRICE LEVELS FOR WHICH DECISION RULES ARE CALCULATED

Current price, 900- to 1100-poundCurrent weight* choice grade slaughter steers

Price range I Representative price Weight range I Representative weight

dollars per cwt pound

20.50-21.50775- 825 800 21.00 21.51-22.50850 22.00826- 875 22 .51-23 .50 23.00876- 925 900 23.51-24.50926- 975 950 24.00

976-1,025 1,000 24.51-25.50 25.00 25.51-26.501,026-1,075 1,050 26.00

1,076-1,125 1, 100 26 .51-27 .50 27.00 1, 126-1, 175 1, 150 27 .51-28 .50 28.00 1, 176-1, 225 1,200

• Average weight of cattle in a particular lot.

of the sell action) for each state of na­ture, we only have to evaluate the possi­ble outcomes of extending the feeding period another 30 days for alternative weight groups and price levels.

The data needed to evaluate the out­comes (equation 12) are presented in t.able 17. Evaluation of equation (12) under alternative price changes consists of three steps: (1) calculate the gross

value at the end of the extended feeding period, (2) calculate current value plus cost of feeding another 30 days, and (3) calculate }I. = GV - [CV+ C].

The average gross value per head at the end of the extended feeding period is defined as

TABLE 17

DATA NEEDED TO ESTIMATE GROSS VALUE OF STEERS AT END OF EXTENDED FEEDING PERIOD AT ALTERNATIVE PRICES

Weight of cattle Proportion of cattle grading Choice at :t

ICurrent 30 days hence•

Cost of feeding another 30 dayst

ICurrent weight End of extended

feeding period

per cent

.263 .411

.360 .472

.432 .523

.490 .566

.538 .602

.580 .636

.616 .666

.648 .690

.677 .715

pounds

800 884 850 933 900 983 950 1,032

1,000 1,081 1,050 1, 130 1, 100 1, 178 1, 150 1, 227 1,200 1,276

dollars/head

18.16 18.76 19.51 20.10 20.63 21.20 21.50 22.01 22.55

• Computed as current weight plus daily gain shown in table 15 times 30. t Cost per pound of gain shown in table 15 times weight gained plus $1.94 = (30)(.0646) nonfeed cost. :t Calculated using equation (11).

21 The decision rules developed here are based on the expected returns per head for a lot of cattle with specified average weight. Therefore, the. prices used to calculate expected returns are a weighted average of Choice and Good grade prices. The proportion of the cattle grading Choice at alternative weights is defined by equation (11).

30 Bii,Uock a1rHJ, Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

. where

W 2 = weight at end of extended feeding period (100 pounds),

r proportion of cattle grad­ing Choice, and

po and pa = price of Choice grade and Good grade slaughter steers, respectively, at the end of the extended feed­ing period (dollars/cwt).

The appropriate values of W2 and r for each weight group are shown in table 17.. Thus, given that the current price is $23.00 and increases $0.50 over the feed­ing period, the gross value at the end of the extended feeding period for a steer currently weighing 850 pounds is

GV = 9.33[.472(23.50) .528(22.52)] = $214.42.

Similarly, given the same price condi­tions, the gross value at the end of the extended feeding period for a steer cur­rently weighing 1,150 pounds is

GV = 12.27[.690(23.20) .310(22.20)] = $280.86.22

The current value of a steer is the gross value of the steer at current weight and prices. For example, the average current value per head of 850-pound steers when the base price is $23.00 is

CV 8.50[.360(23.25) .640(22.36)] = $192.78.

The current value of 1,150-pound steers is

CV= 11.50(.648(22.70) + .352(21.74)] = $257.16.

The cost of feeding steers of alterna­tive weights is shown also in table 17. Thus the outcomes (marginal profit per head) of feeding 850-pound and 1,150­pound steers another 30 days when cur­rent prices are $23.00 and prices in­crease $0.50 over the period are calcu­lated as follows:

850-pound steers: X 214.42 - [192.78 18.76]

= $2.88

and

1,150-pound steers: X = 280.86 - [257.16 + 22.01]

= $1.69.

The outcomes of feeding other weights under alternative price conditions are calculated by the same procedure. For example, the outcomes of extending the feeding period another 30 days for 900­pound steers under alternative price conditions are presented in table 18.

The next step is to calculate the ex­pected profit of each action under al­ternative conditions and select the opti­mal action for each situation.

No data strategies. The expected profit from action Ai is defined as 71'; = ~ ~JP(01). For example, the ex-

i pected profit from feeding 900-pound steers another 30 days when the current price is $23.00 is, 7r .108(14.21) + .117(9.48) .175(4.76) + .217(-.02) .183( -4.74) + .100(-9.47) + .100 (-14.20) = $.24. The expected profits, based on a priori information, for ac­tions A1 and Az relating to 900-pound steers and alternative price levels are shown in table 19. The optimal action (greatest expected profit) for each price level is shown in the last column of table 19. The "no data" strategy for ex­

22 The prices of 1,227-pound steers are keyed to prices of 900- to 1,100-pound Choice grade steers as indicated above. Therefore, the value of the steer at the end of the feeding period is based on a price of $23.50 for Choice grade 900- to 1,100-pound steers.

31 Giann{;ni Foundation Monograph • Nwmber 28 • MOJroh, 1972

TABLE 18 OUTCOMES, X1f, OF EXTENDING THE FEEDING PERIOD 30 DAYS FOR

900-POUND STEERS UNDER ALTERNATIVE PRICE SITUATIONS

Current price

dollars

21.00............. 22.00............ 23.00............ 24.00............ 25.00............ 26.00.......... 27.00........... 28.00............

Price change during extelll!ion of feeding period

01 e. e, 0. e, 01 1.50 1.00 0.50 0.00 -0.50 -1.50

12.44 13.35 14.21 15.02 15.89 16.75 17.62 18.53

7.71 8.57 9.48

10.30 11.16 12.02 12.89 13.80

2.89 3.84 4.76 5.57 6.43 7.30 8.16 9.02

tending the feeding period of 900-pound steers can be summarized as follows: Feed if P ~ $23.00; Sell if P < $23.00, where P refers to the current price of 900-to 1,100-pound Choice grade steers.

The "no data" strategies for other weights of cattle are calculated by the same procedure and are summarized in table 20. For example, the no data strategy is to extend the feeding period of 1,000-pound steers only if current prices exceed $25.00. Generally as the animal's weight increases, higher cur­rent prices are requl.red to continue feed, a factor reflecting increasing costs of gain. Based on the historical distribution of price changes over the 1960-1969

TABLE 19

A PRIORI EXPECTED PROFITS FOR ACTIONS A1 AND A2 AT

ALTERNATIVE PRICE LEVELS: 900-POUND STEERS

Current Expected profit

Optimal actionprice

Ai feed A2 sell

dollars/cwt dollars/head

21.00......... -1.51 0 Sell 22.00......... - .64 0 Sell 23.00......... .24 0 Feed 24.00......... 1.07 0 Feed 25.00......... 1.94 0 Feed 26.00......... 2.81 0 Feed 27.00......... 3.67 0 Feed 28.00......... 4.54 0 Feed

dollars/cwt

-1.74 - .88 - .02

.84 1. 71 2.57 3.43 4.30

-6.47 -5.61 -4.74 -3.88 -3.02 -2.16 -1.29

.43

-11.20 -15.93 -10.34 -15.06 - 9.47 -14.20 - 8.66 -13.39

7.75 -12.48 - 6.88 -11.61

6.02 -10. 75 5.16 - 9.88

period, the expected margin of profit from extending the feeding period of cattle weighing 1,050 pounds is negative regardless of current prices. This result occurs because the weight of the ani­mals 30 days hence will put them in the 1,100- to 1,300-pound price class which typically is lower than the 900- to 1,100­pound category.

The "no data" decision rules sum­marized in table 20 are appropriate if the cattle feeder is willing to base his ex­pectations about future price changes solely on the frequency distribution of these changes in the past. This would be reasonable if (a) no other information about possible price changes were avail-

TABLE 20

BAYESIAN "NO DATA" STRATEGIES, BASED ON A PRIORI

PROBABILITY DISTRIBUTION

Current weight

pound•

800.......... . 850......... .. 900 .......... . 950••.•.......

1,000..... . ... 1,050.......... . 1,100......... .. 1,150......... .. 1,200.......... .

uNo data.11 strategy

Feed regardless of current price Feed if P;;,; 25.00; Sell if P < 25.oo• Feed if P;;,; 23.00; Sell if P < 23.00 Feed if P 2 24.00; Sell if P < 24.00 Feed if P 2"; 25.00; Sell if P < 25.00 Sell regard!""" of current price Feed if P :i1: 27.00; Sell if P < 27.00 Feed ii P;;,; 28.00; Sell if P < 28.00 Sell regardless of current price

• P refers to the current price of 900- to 1,100-pound Choice grade slaughter steers (dollars/cwt).

32 Bullock and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

TABLE 21

EXPECTED PROFIT$ftPER HEAD FROM EXTENDING THE FEEDING PERIOD 30 DAYS FO:it 900-POUND STEERS AT ALTERNATIVE LEVELS

OF PRICES AND PREDICTED PRICE CHANGE

Current price

Predicted price change

z, 1.50

z. 1.00

z, 0.50

z. 0.00

z. -0.50

z, -1.00

Z1 -1.50

dollars

21.00............... 22.00.... .......... 23.00............... 24.00............... 25.00....... " ...... 26.00............... 27.00............... 28.00...............

10.07 10.96 11.84 12.65 13.52 14.38 15.25 16.15

7.71 8.59 9.48

1(),30 11.16 12.02 12.89 13.78

2.11 3.64 4.52 5.36 6.22 7.08 7.95 8.88

dollara/cwt

-2.82 -5.77 -1.96 -4.91 -1.08 -4.04

- .24 -3.20 .63 -2.32

1.49 -1.46 2.36 - .59 3.22 .27

-9.75 -8.89 -8.02 -7.20 -6.30 -5.44 -4.57 -3.71

-14.75 -13.88 -13.02 -12.21 -11.30 -10.43 - 9.57

8.70

able, or (b) the cost of obtaining addi­tional information exceeded its value.

Data strategies. Under the data strategy, profit expectations are condi­tional on the predicted price change. Given an expected price change Zk, the expected profit of action Ai is ~fzk = ~ X.JP(E>J!Zk). P (E>J!Zk) is the kt" column j

of the a posteriori probability distribu­tion presented in table 13.

The expected profit, based on the a posteriori information, from extending the feeding period another 30 days for 900-pound steers when the current price is $21.00 and the predicted price change is $0.50, is calculated as follows: 11'1IZa = .091(12.44) + .273(7.71) + .227(2.98)+ .318(-1.74) + .091(-6.47) = $2.77 Iler head. The matrix of expected profits from extending the feeding period of 900-pound steers for alternative levels of current prices and predicted price changes is presented in table 21.

Since the added profit of action A2 (sell at current weight) is zero, the de­cision rules can be developed solely by

·looking at table 21. Action Ai (extend the feeding period another 30 days) will be the optimal action if l/rli > 0. If

11"1; :::; 0, then A2 is the optimal action. The feed-or-sell decision rules for 900­pound steers presented in table 22 are derived following these criteria.

The la.st column of table 22 shows the minimum predicted price change (/3,.) 23

required to extend the feeding period at alternative levels of current price (i.e., the minimum value of Zk for which '?? > 0). The decision rule is to extend the feeding period if Zk ~ /3k and to sell at current weight if z,, < /3k. For ex­ample, if the current price is $23.00, a

TABLE 22

BAYESIAN "DATA" STRATEGIES, BASED ON A POSTERIORI

INFORMATION: 900-POUND STEERS

Minimum predicted

Current "Data" strategy, given predieted priceprice price change Z• change

required to feed <ft•)

dollars/cwt

21.00.... Feed if Zk '?; .50; Sell otherwise .50 22.00.... Feed if z. ~ .50; Sell otherwise .50 23 .00.... Feed if z. ;2; .50; Sell otherwise .50 24.00.. ,. Feed if Zk ~ .50; Sell otherwise .50 25.00.... Feed if z. <= .00; Sell otherwise .00 26.00.. , Feed if z. ~ .00; Sell otherwise .00 27.00.... Feed if Zh «; .00; Sell otherwise .00 28.00.... FeedifZ• «;-.50;Sellotherwise - ..50

2a f3k changelil in intervals of $0.50 because expected profits were evaluated for $0.50 intervals in predicted price change zj;.

33 Giannini Foundation Monograph • Number 28 • March, 1972

TABLE 23 SUMMARY OF BAYESIAN "DATA" DECISION RULES FOR SHORT-RUN

MARKETING DECISIONS-MINIMUM PREDICTED PRICE CHANGE REQUIRED TO EXTEND FEEDING PERIOD 30 DAYS FOR

ALTERNATIVE WEIGHTS OF CATTLE AND CURRENT PRICES

Current weight of cattle

Current price 775­825 I

826­875 I

876­925 I

926­975 I

976­1025 I

1026­1075 I

1076­1125 I

1126­1175 I

1176­1225

dollars/cwt

20.51-21.50............. 21.51-22 .50 ............. 22.51-23.50 ............. 23 .51-24 .50 ............. 24.51-25.50............. 25 .51-26.50 ............. 26.51-27.50 ............. 27.51-28.50 .............

.50 0 0 0

-.50* -.50 -.50 -.50

.50

.50

.50

.50

.50 0 0

-.50

.50

.50

.50

.50 0 0 0

-.50

.50

.50

.50

.50

.50 0 0 0

.50

.50

.50

.50

.50

.50 0 0

1.00 1.00 1.00

.50

.50

.50

.50

.50

.50

.50

.50

.50

.50

.50

.50

.50

1.00 .50 .50 .50 .50 .50 .50 .50

1.00 .50 .50 .50 .50 .50 .50 .50

•A negative value indicates that the expected profit from continued feeding is positive even if a price decline of this magnitude is expected.

predicted price change zk ~ $0.50 is re­quired to extend the feeding period for 900-pound steers.

The Bayesian strategies for other weights of cattle are derived by the same procedure and are summarized in table 23. The values shown in the table are the minimum predicted price change required to extend the feeding period for each weight category at alternative levels of current prices. For example, if we have a lot of steers averaging 1,200 pounds and current prices are $21.35, a predicted price change Z ~ $1.00 is required to extend the feeding period. If current prices are $26.15, we can expect positive marginal profits from extending the feeding period of these steers only if the predicted price change zk ~ $0.50.

The above decision rules are consistent with a priori expectations, given that the cost per pound of gain increases with the weight of the steer. The critical values of table 23 (i.e., the minimum expected price change required to ex­tend the feeding period) tend to de­crease as prices increase and increase as current weight increases.

Value of the price forecasting model. The value of the information provided by the price forecasting model is defined by equation 4 as the difference between the expected income from fol­lowing the data strategy and the ex­pected income from following the no data strategy. Consider, for example, the value of the data for 900-pound steers at alternative price levels.24

The no data strategy for 900-pound steers is to extend the feeding period if the current price is greater than or equal to $23.00 and to sell at current weight otherwise (table 20). The expected in­come from following the no data strat­egy is the expected profit (table 19) of the optimal action at each price level. For example, the expected (marginal) income from following the no data strategy (i.e., sell) is zero when the price is $22.00 and $2.81 per head when the price is $26.00.

The data strategy (table 23) for 900­pound steers is to extend the feeding period if (a) p ::; $24.00 and zk ~ $0.50, (b) $27.00 ~ p > $24.00 and zk ~ 0, or P ~ $28.00 and Zk ~ -$0.50, and to

24 Since each price level is considered as a separate problem, the value of the data is com­puted for each price level.

34 Bulloak and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

TABLE 24

VALUE OF INFORMATION PROVIDED BY THE PRICE FORECASTING

MODEL: 900-POUND STEERS

Expeeted income from following each strategy

Current Value of price data

Data No data strategy strategy

dollars/cwt dollars/head

20. 50-21. 50......... 2.06 0.00 2.06 21.51-22.50......... 2.39 0.00 2.39 22.51-23.50......... 2.72 .24 2.48 23.51-24.50......... 3.02 1.07 1.95 24.51-25.50......... 3.51 1.94 1.57 25.51-26.50......... 4.05 2.81 1.24 26.51-27.50......... 4.60 3.67 .93 27 .51-28.50......... 5.22 4.54 .68 Average............ 3.45 1.78 1.67

sell at current weight otherwise. Because the expected profit from extending the feeding period is conditional on the pre­dicted price change, it is necessary to weight the expected profits for alterna­tive values of zk (table 21) by the probability of predicting Zk, P(Zk) (ta­ble 11). Thus, the expected (marginal) income, i, from following the data strategy for 900 pound animals when P = $22.00 is

i = .050(10.96) + .136(8.59) +.185(3.64) + .226(0) +.108(0) + .034(0)

= $2.39

t The expected income from following each strategy for 900-pound steers is presented in table 24. The value of the data ranges from $2.48 when P = $23.00 to $0.69 when P = $28.00. The value of the data strategy over the alternative method for 900-pound steers averages $1.67 per head.

During the 1960-1969 period, the price of 900- to 1,100-pound Choice grade steers averaged about $25.00. At

this price level the expected (marginal) income per head over all weight groups from incorporating the information pro­vided by the price forecasting model into the decision process exceeds by $2.00 per head the expected (marginal) income of basing the decision solely on past price changes.

Sensitivity of model. The decision rules developed above are based on esti­mates of costs and returns under al­ternative situations for a "typical" Cali­fornia feedlot. This procedure raises two questions. First, how sensitive is the model to errors made in estimating costs? Second, are the decision rules ap­plicable to feedlots whose cost relation­ships deviate from those specified for the typical feedlot? The second question is answered by answering the first. If the same decision rules would arise for a range of cost estimates, then we can conclude that the decision rules are ap­plicable for feedlots with atypical cost relationships.

We can evaluate these questions by looking at the expected profit from ex­tending the feeding period, given zk = ek and zk = [Ilk - $0.50] at alternative price levels.25 Consider for example, the case of 900-pound steers presented in table 25. The values in column 3 indi­cate the expected profit from extending the feeding period of 900-pound steers given Zk = fJk. They also indicate the amount by which estimated costs per head would have to be increased before fJk would be increased to the next high­est value. The values in column 4 indi­cate the amount by which cost esti­mates would have to be decreased be­fore fJk could be reduced to the next lower value. Taken together, these fig­ures specify the range of cost estimates for which the decision rules are appli­

20 Zk is the expected price change, measured in $0.50 increments. fJk is the minimum value of Z required to extend the feeding period (i.e., the minimum expected price change for which ex~ pected profit of continued feeding is positive).

35 Giannini Foundation Monograph • Number 28 • March, 1972

TABLE 25

DATA FOR EVALUATING SENSITIVITY OF DECISION MODEL:

900-POUND STEERS

Minimum price Current change required

price to extend feeding

Expected profit from feeding at alternative

values of Zk

period (f3k) z. = /3• Iz. = /3• _ $0.50

dollars/cwt dollars/head

21.00..... .50 2. 77 -2.82 22.00.. ". .50 3.64 -1.96 23.00.. ". .50 4.52 -1.08 24.00..... .50 5.36 - .24 25.00.. ". 0 .63 -2.32 26.00.. ". 0 l.49 -1.46 27 .00.. ". 0 2.36 - .59 28.00.. ". -.50 .27 -3.71

cable. For example, given P = $21.00, we would arrive at the same critical value, f3k = $0.50 for estimated cost of feeding ranging from $22.28 to $16.69.26

Note that f3k changes from $0.50 to .00 as we move from P = $24.00 to $25.00. We will refer to P = $25.00 as the pivot point (the price at which f3k

changes value). The location of the pivot point is sensitive to cost estimates. In the above example, if feeding costs per head had been 25 cents lower than the estimate shown in table 17, the pivot point would have been P = $24.00. On the other hand, if feeding costs would have been 64 cents higher than the esti­mate used, the pivot point would have been P = $26.00. Thus, if the decision rules are being used for a feedlot whose costs are above those estimated here, the pivot point could be moved from P = $25.00 to P = $26.00. Similarly, the pivot point could be moved to P = $24.00 if the cost structure is below that used above.·

Based on this analysis, the decision rules are rather insensitive to variations

of feeding costs and are applicable to a wide range of feedlot situations. How­ever, as noted above, a slight modifica­tion of the pivot point might be in order if costs of feeding deviate from the esti­mates used here. Evaluation of similar data for other weight groups supports these conclusions.

Part of the stability of the decision rules is built into the model by using $0.50 increments in price changes. Smaller price change increments would make the model more sensitive to varia­tion in per head feeding cost. The deci­sion rules summarized in table 23 are conservative since there may be pre­dicted price changes less than f3k for which the expected marginal profit from continued feeding is still positive. All values of predicted price changes be­tween (f3k - $0.50) and f3k were not evaluated since our probability distribu­tions were geared to $0.50 increments in price change. Therefore, f3k may be an overestimate of the breakeven predicted price (i.e., the predicted price change for which fr= O).

Define the break-even price change, {3fo, as {3{, = {3k - ~/W2, where '11- is the expected per head marginal profit if Zk = f3k and W2 is the weight of the steer (100 pounds) at the end of the ex­tended feeding period. There is a lower bound on {3fo, defined by the lower bound of the price change interval for which f3k = zk is the representative value.27

Thus, if f3k = Zk = $0.50, the lower bound of {3{, is $0.26 since the probabili­ties used to calculate the expected in­come for Zk = f3k are applicable only to price changes in the range $0.26 =:::; Zk =:::; $0.75. Consequently, a different set of probabilities has to be used to calculate expected income if Zk < $0.26. There­

26 Estimated cost of feeding 900-pound steers another 30 days is $19.51. $19.51 + $2.77 = $22.28 and $19.51 - $2.82 = $16.69.

27 The intervals for which z. is the representative price are the same as the intervals for 0;, j = k as shown in table 7.

36 Bullock and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

TABLE 26

EXPECTED PROFIT, .r, (DOLLARS PER HEAD) FROM EXTENDING THE FEEDING PERIOD ONE MONTH: GIVEN THAT Zk Pk

Weight of cattle on feed (pounds)

Current price of slaughter steers

dollars/cwt

20.50-21.50............. 21.51-22.50............ 22.51-23.50............ 23.51-24.50..... .... 24.51-25.50............. 25.51-26.50... , ....... 26.51-27.50............ 27.51-28.50............

876­775­ 826­825 875 925

926­ 976­ 1,026­ 1,076­ 1, 126­ 1, 176­975 1,025 1,075 1,125 l, 175 1,225

4.12 .21

1.04 1.92

.27 1.11 1.98 2.81

.26 1.48 2.66 3.89 5.10

.98 2.18

.56

2.77 3.64 4.52 5.36

.63 1.49 2.36

.27

2.16 3.02 3.89 4.71 5.56

.51 1.36 2.22

dollars

1.63 2.47 3.32 4.13 4.97 5.80

.45 1.29

3.18 4.14 5.12

.26 1.27 2.27 3.26 4.27

.41 6.24 6.01 1.21 .69 .16 2.00 1.45 .92 2.86 2.30 1.77 3.68 3.11 2.57 4.49 3.91 3.35 5.34 4.75 4.18 6.16 5.55 4.98

fore it is necessary to set a lower bound on f3l equal to the lower bound on Z11; = f3k·

The decision rules can be modified with the information about {3£,. For ex­ample, if the decision rules specify f3k = $0.50 and it turns out that f3t = $0.30, then if zk > $0.30, the expected profits from extending the feeding period are positive. In this case, the cattle feeder might want to extend the feeding period even though a sell decision is generated by the decision rules of Model I.

The above modification of the de­cision rules emphasizes the accuracy of the estimated cost of feeding another 30 days. This modification is applicable only if the cost per head of feeding an­other 30 days in the feedlot for which t}ie decision is being made (C') is identi­cal with the estimated cost ( C) used to derive the decision rules. If C' ~ C, then the appropriate adjustment can be calculated as f3J,' = f3k - [11' - (C' - C)J/ W2• Thus, for feedlots where C' > C, f3!/ will be less than f31' and if C' < C, then {3~1 will be greater than f3f,. The same set of lower bounds applies for f3!/ as does for f3l· The values of 1r needed to make these adjustments are shown in table 26.

The expected income of the strategies could be increased if decisions were

based on a comparison of Z11; with f3f, rather than f31c· However, these decision rules would be incorrect for feedlots with per head feeding costs above those used in the study and would still be con­servative for feedlots whose costs are below the estimates used. Therefore if a more precise decision rule is desired, the values of f3k in table 23 would have to be replaced by the values of {3~1 which are applicable to the feedlot in question.

Feeding decisions for two monthS-­Model II

The purpose of Model II is to develop a set of decision rules utilizing 60-day price predictions with which to re­evaluate a sell decision generated by Model I for steers weighing less than 1,000 pounds.

Model II uses the same theoretical framework and similar input data as Model I with a reduced number of states of nature (price changes) from seven to three. The states of nature used in Model II were shown previously, and the a posteriori probability distributions for 60-day price projections are pre­sented in table 12.

The weight of cattle at the end of the extended feeding period, cost of feeding, and proportion grading Choice at alter­native weights are presented in table 27.

37 Giannini Foundation Monograph • Nwmber 28 • March, 1972

TABLE 27 INPUT DATA NEEDED TO DERIVE . DECISION RULES FOR MODEL II

Weight at Proportion of cattleend of grading choice atCost of

extended feedingCurrent fee~inJ'weight per10

End of 60-dayCurrent

60 days weight60-day extension

pounds dollars/head

37.67 .263 .508800 967 850 1,015 38.86 .360 .551

40.14 .432 .590900 1,064 .490 .624950 1,112 41.31

Given this infonnation along with the a posteriori distributions, the decision rules are calculated by the same procedure as used for Model I.

The resulting decision rules are sum­marized in table 28. The values shown in the table are the minimum expected price change ((:Ji.) required to extend the feeding period. If the predicted price changes are below the values in the table the cattle are sold. For example, given a lot of steers averaging 850 pounds and the current price of $23.20, a predicted price change of $1.00 over the next 60 days is required to override a sell de­cision in Model I (table 28). If the pre­dicted price change over the 60-day period is less than $1.00, the decision is to sell.

Feeder cattle purchase decisions­Model III

Model III develops decision criteria for purchasing feeder cattle. The model specifies that the cattle feeder purchases 600-pound Good grade feeder steers, based on the expected value of the steer at the end of a 150-day feeding period.28

The decision rules compare the pre­dicted price of Choice grade 900- to 1,100-pound slaughter steers five months

hence with the break-even price of slaughter cattle, given the current price of feeder cattle. The break-even price (P8 ) is the price of a 900- to 1,100­pound Choice grade slaughter steer re­quired at the end of the feeding period to equate gross value of the slaughter steer with the cost of the feeder steer plus cost of feeding. Fed cattle weighing 1,015 pounds on the average will grade 55.l per cent Choice and 44.9 per cent Good (equation 11). Given the relation­ship between the price of Good grade slaughter steers and that of the Choice grade animals (table 5), the required breakeven gross value of the steer at the end of the feeding period can be defined as: GV = 10.15 [.551P8 + .449(.848 + .922P8 )], or GV = 3.865 9.794P8 •

The cost of feeding over the 150-day period has two components, feed costs, and nonfeed costs. Feed costs per pound of gain average 19.31 cents over the 600­to 1,000-pound feeding interval (table 15). Thus, the feed costs over the 150­day feeding period are (415)(.1931) = $80.14. Nonfeed costs total (150)(.0646)

TABLE 28

SUMMARY OF DECISION RULES FOR MODEL II-MINIMUM PREDICTED

PRICE CHANGE OVER NEXT 60 DAYS REQUIRED TO OVERRIDE A SELL

DECISION FROM MODEL I

Current price

Current weight of steers

775-825 826-875 876-925 926-975

dollars/cwt

20.51-21.50... 1.00 1.00 1.00 1.00 21.51-22.50... 1.00 1.00 1.00 1.00 22.51-23 .50... 1.00 1.00 1.00 1.00 23.51-24.50... 0 LOO 1.00 1.00 24.51-25.50 ... 0 0 0 LOO 25 .51-26.50... 0 0 0 LOO 26.51-27.50... -1.00* 0 0 1.00 27 .51-28.50... -1.00 -1.00 0 0

•Negative values indicate that continued feedingwould be profitable even if prices declined by $1 .00/cwt.

2sThis- corresponds to the "normal" feeding period of 600--1,000 pounds used previously. If the cattle gain 2.77 pounds per head per day (table 15), they will weigh 1,015 pounds at the end of the 150-day feeding period.

38 Bullock and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

= $9.69 per head. Therefore, total cost of feeding a steer for 150 days is esti­mated to be $89.83.

The break-even price of slaughter cattle (PS) can be calculated for alterna­tive levels of current feeder cattle prices (PF) as shown by equation (13).

10.15 [.551PS + .449(.848 + .922PS)j = 6.0PF = 89.83

3.865 + 9.794PS = 6.0PF + 89.83 (13)

ps = 8.777 + .613PF

If the current price of feeder cattle29 is $24.00 per hundred pounds, the break­even price of Choice grade slaughter steers five months hence is ps = 8.777 + .613(24.00) = 23.49.80

For a given level of current feeder cattle prices, the cattle feeder will pur­chase feeder cattle only if the predicted price of 900- to 1,100-pound Choice grade slaughter steers is above the break-even price. That is, feeder cattle will be pur~ha§.tld only if the expected profit margm M = P 6- ps > 0, where P6 is the predicted price of slaughter cattle five months hence.

The cattle feeder is concerned about the risk of loss involved as ·well as the expected profit margin. That is, given an expected profit margin, M= (Ps ~ P'j), he is concerned about the probabil­ity that P6 - ps < O, where P. is the realized price at the end of the feeding period.

The price predictor developed earlier by least-squares estimation assumes that the error term associated with the equa­tion is normally distributed with mean

TABLE 29

POSSIBLE DECISION RULES FOR PURCHASING FEEDER CATTLE

BASED ON COMPARISON OF EXPECTED PRICE, P., AND

BREAK-EVEN PRICE ps

Expected profit PropoJ:iion of Probability of a. ,,., margin ,,.,. vacant capacity loss Prob Po-PS M to be purchased !Po - ps < O[MJ

per centdollars/cwt

No purchase,ft°" .25 .26:s:M:s: .75 .15 < 48 .76:s:M:s:us .30 < 30

.45i.2a"' tr o;; i.75 <20

.60i.1a;,; M."' 2.2s < 10 2.26 o;; tr .75 < 5

0 and variance f1' 2• On this basis, the variable P5 - PsfS·P6 follows a t-distribu­tion with n - k degrees of freedom (Goldberger, 1964, p. 179),31 where Sn is the standard error of the estimate. Consequently, the prediction errors (P5 - P5) are symmetric about zero. By restricting purchases to situations M = (P5 - ps) > 0, the probability of a loss occurring (i.e., Prob {(P6 - PS) < O}) never exceeds 50 per cent. Moreover, as ,... M increases, the probability of a loss decreases.32

The decision rules developed below are based on three hypotheses. First, some minimum expected profit margin, M> 0, is necessary to induce the cattle feeder to place cattle on feed, depending on the cattle feeder's aversion to risk. Second, the number of cattle placed on feed at any given time is a function of the expected profit margin, given the vacant capacity in the feedlot. For ex­ample, more cattle would be placed on feed if M= $3.00 than if M= $0.50. Third, placements of cattle on feed in any given month will be less than 100

'9The price referred to here is the price of 550- to 750-pound Good grade feeder steers. 30 The break-even margin is calculated as the break-even price minus the current price of

feeder cattle, i.e., $23.49 - $24.00 = $-0.51. n n is the number of observations used to estimate the equation and k = the number of pa-­

rameters estimated in the equation; n = 38 and k = 11 for the first two five-month price pre­diction equations shown in table 11 and n = 39 and k = 11 for the third equation.

39 GianniniFoundaUQ11,Monograph • Nu,mber 28 • March, 1972

per cent of vacant capacity, and there is a limit to the number of feeder cattle that will be purchased in any one month (e.g., 20 per cent of total feedlot ca­pacity).

The decision rules presented in table 29 are admittedly arbitrary and will vary from feeder to feeder depending on this aversion to risk and the operating characteristics of the feedlot. The mini­mum expected profit margin required for purchasing feeder cattle (i.e., $0.25) and the proportion of vacant capacity to be filled at alternative values of M are not based on any particular opti­mizing criteria; rather, they are pre­sented as an example of the decision rules that can be developed. The value of these parameters can readily be changed to conform with the desires of a given cattle feeder. The only calcula­tions required to make these changes are the probabilities of making a loss (i.e., Prob {Pi; - ps < OjPi; - M = PB}).

The probablity of a loss given a pre­dicted profit margin, M= Po - PB, is calculated as follows:

Prob {P5 - PB < OjPi; - M= P 8 }

=Prob {Pa - P6 < -M}

= P b P6 - P6 < -ii ro SA P5

-M=Probt<F . p.

The probability that t < -M/SP6 can be found from a table of values for the t-distribution using (n - k) degrees of freedom.

Since there are separate equations for making five-month forecasts for the three months of the quarter, there is one value of S.Po($1.28) the first month of the quarter, another 8.Pi;($1.38) the sec­ond month, and still another S.P5($1.55) for the third month of the quarter. The probability of a loss, given Pi; - M = P 8 , differs only slightly between these values of S;.o but increases as Sp5 in­creases.

The break-even slaughter cattle price equation can be adjusted by replacing the estimated feeding cost (C) with the actual cost ( C') of the particular feedlot. The minimum expected profit margin M required for purchasing feeder cattle can be specified by the cattle feeder de· pending on his aversion to risk. The quantity purchased criteria in column 2 of table 29 also will depend on the operating characteristics of the feedlot.

Once the cattle feeder has specified the appropriate parameters, only a few simple calculations are needed to make purchasing decisions. First, calculate the break-even price of slaughter cattle five months ahead, given the current price of feeder cattle. Second, project the price of slaughter cattle five months ahead. Third, calculate the expected profit margin M = P6 - PB. And, fourth, using the decision parameters specified by the feedlot operator, de­

a2 This is shown as follows: Prob {Ps < PSJP. - M=PS) = Prob {Pi;< Ps - M}

=Prob IPs -Ps <-Ml P.-P. M

= Prob-s-- < ­P5

M= Probt <-­BP5

Given that M1 > M2, we know,

Prob{t < - ~}<Prob {t< -!:}·

40 BulZoo'k and Logan: Cattle Feedlot Marketing Decisions Under Uncertainty

termine the number of head to be pur­chased. The probability of loss occurring ...... (i.e., P 5 < P 8 ) given M can be calcu­lated as shown above.

Purchase and sales decisions-:­Model IV

Uncertainty about f,uture prices pre­cludes precise planning as to the pattern of placements and marketings into the future; however, some tentative plans are needed about future sales and pur­chases in order to assess future needs for capital and feed. Model IV incorporates the decision rules developed, along with the results of the price forecasting model, into a six-month planning model. The resulting information can only be con­sidered as tentative and is not precise enough to serve as a basis for definite decisions, but does provide an indication of the possible pattern of sales and pur­chases over the next six months and thus, may be useful in forward planning.

Model IV simulates the buying and selling activities of the feedlot over the next six months, given the current in­ventories of cattle in the feedlot and the capacity of that lot. This simulation is based on predictions of slaughter cattle prices up to 11 months ahead and feeder cattle prices up to five months ahead. The decision rules developed are applied sequentially to the marketing and pur­chasing decisions that would confront the cattle feeder over the next six months if the predicted prices were in fact the true prices. The resulting pattern of pur­chases and sales serves as a tentative identification of the set of decisions that may evolve during the next six months.

Given the above information, the model (1) uses the results of Models I

and II to calculate sales for the first month (i.e., decide which lots currently on feed should be sold and which lots continue feeding), (2) calculates vacant capacity as total capacity minus in­ventory less sales, (3) decides how many feeder cattle will he purchased in first month, based on decision rules and pre­dicted prices in Model III; and (4) calcu­lates inventory of .cattle on feed by weight groups at the end of the first period. The inventories then become be­ginning inventories for period 2, and the four steps are repeated for the second period. This process is continued through six months. At the end of the sixth sequence, one can observe the expected pattern of sales and purchases.

As an example of how Model IV is used, consider the following tentative plan of sales and purchases over a six­month period for a 10,000-head capacity feedlot with the following current in­ventory of cattle on feed:

Weight33 Number currently pounds in feedlot

W1 600 500 W2 = 684 1,500 W3= 767 500 w4 850 1,500 w6 = 933 500 w6 = 1,015 1,500 W1 = 1,096 500 Wa = 1,174 1,000

w9 1,250 500

Total 8,000

In addition, we have the following price information and predictions.34

33 All cattle are assumed to be placed on feed weighing 600 pounds. The daily gain information presented in table 15 defines the above weights for 30-day feeding intervals. That is, cattle on feed 60 days will weigh 767 pounds, and so on.

34The i subscript on P and PP refers to the length of projection. Thus, P. is the projected slaughter price five montpa hence: i = 0 indicates current prices.

41 Gianwini Foundation Monograph • Number 28 • March, 1972

Price of 900- to 1,100-pound Price of 550- to 750-pound Choice grade steers Good grade feeder steers

doll,ars per hundredweight

Po= 24.95 Ps = 23.93 PPo = 23.05 P1 = 25.78 P1 = 25.74 PP1 = 23.50 P2 = 25.03 Ps = 25.09 PP2 = 23.50 Pa= 24.44 p9 = 24.28 PPa = 23.50 p4 = 25.18 Pio= 25.05 pp4 = 23.38 p5 = 24.23 Pu= 24.30 pp5 = 22.26

The first step in the simulation is to determine the number of cattle that will be sold in the first month. In accordance with the decision rules of Model I, cattle weighing less than 775 pounds (i.e., the cattle in weight groups W1,

W2, and Wa) will be continued on feed and cattle weighing more than 1,225 pounds (weight group W9) will be sold. The decision to keep or sell the other cattle is based on the decision rules summarized in table 24. The predicted price change Zk = P1 - Po = $0.83. Since this exceeds the minimum value specified in table 23 for each weight group given the current price is $24.95, the decision in period one is to extend the feeding period for the cattle in weight groups W4, Ws, Ws, W1, and Ws. Thus, sales in period one equal 500 head from weight group w9.

The second step is to calculate vacant capacity in the feedlot after the sales indicated above. Sales in period one are 500 head. Thus vacant capacity =

10,000 - (8,000 - 500) = 2,500 head. The third step is to calculate the

number of cattle to purchase in period one based on the purchasing decision rules presented in table 29. The current price of feeder steers in the first period is $23.05 thus the break-even price of

P 8slaughter cattle = 8.777 + .6.13(23.05) = $22.91. The expected prof­it margin M = P5 - P 8 = 24.23 ­

22.91 = $1.32. The decision rules in table 29 indicate that for 1.26 :::; M :::; 1.75 purchases of feeder cattle will be 0.45 (vacant capacity) = 0.45(2,500) =

1,125 head. We can now summarize the sales and

purchase transactions for period one and calculate the inventories for the be­ginning of period two. Five hundred head of 1,250-pound steers were sold in period one and 1,125 head of 600-pound steers were purchased. Retained cattle on feed from the beginning of the period will advance one weight group by the beginning of the second period (neglect­ing death loss). Thus, the beginning in­ventory for the second period is as follows:

W1 = 1,125 w4 = 500 W1 = 1,500 W2 = 500 w5 = 1,500 Ws = 500 Wa = 1,500 Ws = 500 w9 = 1,000

The predicted price change between the first and second period is Z = $25.03 - $25.78 = $-0.75.35 Thus, based on the decision rules summarized in table 23, the decision is to sell all cat­tle in weight groups W4, W5, Ws, W1, and W8• As in period one, cattle in weight groups W1, W2, and Wa will be kept and cattle in weight group W9 will be sold. The cattle in weight groups W4 and w5, however, are less than 1,000 pounds; therefore, it is necessary to

35 P 1 takes on the role of current price since the decision rules are being evaluated at the begin­ning of period two (end of period one).

42 BuZwclc and Logan: Cattle Feedlot Marlceting Decisions Under Uncertainty

TABLE 30

SUMMARY OF SIMULATING MARKETING AND PURCHASING DECISIONS SIX MONTHS INTO THE FUTURE FOR HYPOTHETICAL EXAMPLE

Sales

Period= 0 Period 1

500

Period 2

5,500

Purchases 1, 125 1,031

Weight group

evaluate the expected returns from ex­tending the feeding period 60 days based on rules developed in Model II.

The predicted price change over 60 days, is $-1.34. The decision rules summarized in table 28 indicate that for P 1 = 25.78 and cattle weighing 850 and 933 pounds, a predicted price change of zero or greater is required to extend the feeding period beyond 30 days for the 850-pound animals and $1.00 or more for the 933 pound animals. Therefore, the sell decision generated above is ap­propriate, and sales for period two are 5,500 head.

Vacant capacity for the second period i~ 10,000 - (8,625 - 5,500) = 6,875.

The current price of feeder cattle in period two is $23.50. Therefore, the break-even price is ps 8.777 + .613(23.50) = $23.18. The predicted slaughter price five months hence is $23.93, thus M= $0.75. The decision rules in table 29 indicate that when ..... 0.25::;; M::;; 0.75 purchases of feeder cattle = 0.15 (vacant capacity) = .15(6,875) = 1,031 head.

Thus sales= 5,500 head in period two and purchases= 1,031 head. The

Period 3

1,500

Period 4

0

Period5

500

Period 6

0

2,000 2,000 l, 153 2,000

Inventories

W1 .......... ··········· w....................... w,. .................... w....... .......... ..... w, .. ....... ...... ... w....... ··············· W1··············· .. ·· " Wo .. ..... .. .... ...... w........... . . . . . . . . ' . .

Total. .................

500 1,500

500 1,500

500 1,500

500 1,000

500

8,000

1,125 500

1,500 500

1,500 500

1,500 500

1,000

8,625

1,031 1, 125

500 1,500

0 0 0 0 0

4,156

2,000 1,031 1,125

500 0 0 0 0 0

6,656

2,000 1, 153 2,000 2,000 2,000 1, 153 1,031 2,000 2,000 1, 125 1,031 2,000

500 1, 125 1,031 0 0 1, 125 0 0 0 0 0 0 0 0 0

7,309 7,309 9,309

inventories at the beginning of period three are as follows:

W1 = 1,031 W2 1,125 w. 500

The procedure used to simulate pe­riods one and two is continued through periods three, four, five, and six. The results of Model IV for the above prob­lem are summarized in table 30.

Model IV is a planning model only in the sense that it may provide some in­sight as to future developments. The simulation procedure of Model IV does not develop a set of decision rules by which future marketing and purchasing decisions can be made. Rather, it pro­vides only an indication of the sales and purchasing pattern that may evolve over the next six months. In order to use the model for a particular feedlot, one would only have to change the capacity constraint to the appropriate level and replace the feeder cattle purchasing cri­teria with those specified by the cattle feeder.

43 Giannini Foundation Monograph • Number 28 • March, 19'72

SUMMARY AND CONCLUSIONS The purpose of this study has been to

present an application of statistical de­cision theory to managerial decision­making in cattle feedlot procurement and marketing.

The analysis indicates the value of more sophisticated decision-making mechanisms to business executives. Ad­ditional information through price fore­casting or other experiments can help increase the expected returns to a firm and reduce the variance of outcomes from such decisions. If a firm is vulner­able to large losses, reduction in risk and uncertainty may become a major goal in itself in decision making. Thus, a means of incorporating uncertainty directly in­to the decision process should prove useful.

In this analysis, the primary focus was on the uncertainty associated with imperfect knowledge about prices of fed cattle. An attempt was made to reduce the uncertainty surrounding marketing and procurement decisions by utilizing a price forecasting model in the decision process. The results of the marketing model indicate that use of such an ex­plicit price predicting method would add about $2 per head to the expected rev­enues as compared with a more naive marketing strategy.

The price forecasting model projects average monthly prices of 900- to 1,100­pound Choice grade slaughter steers at El Centro, California, up to 12 months ahead. Similar projections are made for prices of 550- to 750-pound Good grade feeder steers at Stockton, California, up to six months ahead. The model is not a perfect price predictor, but it does pro­vide a means whereby the cattle feeder can utilize information about future price changes in making decisions. The conditional probabilities of actual price

changes attaining certain magnitudes given the predicted price change permits tlie feeder to make decisions as to whether to continue feeding animals cur­rently in the feedlot an additional month or two or whether to sell them at their current weight.

While the models in this study illus­trate the use of statistical decision theory, they are simplified to some de­gree and may require adjustment be­fore actual application to a given feedlot operation. For example, factors such as weight gain, feed conversion, or change in grade affect costs or revenues. These factors all have stochastic attributes which have been considered negligible by setting their levels at their expected values. To the extent that these varia­bles assume values more detrimental to profits than postulated here, the net effect of this type of uncertainty would be reflected in a greater needed expected price increase than suggested by the model before reaching the decision to feed another month. A sensitivity check of the models, however, shows that similar results would be obtained over a fairly wide range of costs.

Another limitation may be found in the one-month time period allocated to the decision process. Actual decision periods are continuous, not discrete. However, the nature of the data avail­able for the price forecasting model re­stricted the time period to monthly in­tervals. This limitation, however, is not critical to the nature of the decision model, since the same procedure could be used for any length of time.

Because the replacement component of decision making has not been included, the models are more generally applicable to feedlots operating with excess capac­ity and available capital for additional

44 Bullock 1111td Logan: Cattle Feedlot Marketing Deewions Under Uncertainty

investment in feeder cattle. Similarly, the models can be used by small feedlots which feed only a single lot per year.

Probably the greatest area for po­tential improvement, however, is in the price forecasting system. Modification of the model could include explicit rec­ognition of the effects of prices or quan­tities of other meat animals and of changes in income. A tendency toward lighter marketing weights observed in 1967-1968 suggests that the average weight of cattle sold from feedlots may be an important variable. The model used in this study is based only on num­bers of cattle marketed and lagged prices, and therefore, assumes that the average weight of fed cattle marketed will remain at the 1960-1969 average. Consequently, if average weights de­cline (increase) the model will over (under) estimate the supply of beef actually coming on the market and thus under (over) estimate prices.

The problem of making the modifi­cations necessary to adjust for the above factors will be the ability to predict significant variations in these variables and still maintain the simplicity of the model. A balance will have to be at­tained between the value of the informa­tion gained as reflected in improved

prediction accuracy and the cost of the added information in terms of collection costs and increased complexity.

No attempt was made in this study to compare the decision rules developed with other decision rules. Comparison of expected incomes arising from the de­cision rules developed in the study with decision rules used by actual cattle feeders would be of particular interest, provided one could explicitly state the decision rules used by a cattle feeder. The comparison of average income and income variability could be accom­plished by simulating the operations of the feedlot over some period of time under these alternative decision rules. For example, one could simulate the operations and resulting income flows using (a) the "data" strategies, (b) the "no data" strategies, (c) perfect knowl­edge, and (d) other decision rules avail­able to feedlot operators such as mar­keting at 1,000 pounds regardless of current prices. Information gained from this simulation would provide a basis for selecting among alternative decision rules by considering income variability as well as expected income. The survey and quantification of decision rules used by cattle feeders constitutes a study in itself and thus was not attempted here.

LITERAWRE CITED BERKSON, J. ' 1950. Are there two regressions? Jour. of Amer. Statistical Assoc., 45:164---80. BULLOCK, J.B.

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CHERNOFF, H. 1954. Rational selection of decision functions, Econometrica, 22:422-43.

CHERNOFF, H. and L. E. MosEs 1959. Elementary decision theory, New York: John Wiley and Sons, Inc.

DEAN, G. W., A. J. FINCH and J. A. PETIT, JR. 1966. Economic strategies for foothill cattle ranches, University of California Agr. Expt.

Sta. Bui. 824. DRAPER, N. R. and H. SMITH

1968. Applied regression analysis, New York: John Wiley and Sons, Inc.

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EHRICH, R. L. 1966. Economic analysis of the United States beef cattle cycle, University of Wyoming Agr.

Expt. Sta. Science Monograph 1.

EIDMAN, VERNON R., HAROLD 0. CARTER and GERALD W. DEAN 1968. Decision models for California turkey growers, University of California, Giannini

Foundation Monograph 21.

FARIS, .J.E. 1960. Analytical techniques used in determining the optimal replacement pattern, Jour. of

Farm Economics, 42(4):755-66. FOOTE, R. J.

1958. Analytical tools for studying demand and price structures, U.S. Department of Agri­culture, Agr. Handbook 146.

GARRETT, W. H., J. H. MEYER and G. P. LOFGREEN 1959. The comparative energy requirements for sheep and cattle for maintenance and gain,

Jour. of Animal Science, 18(2):528-47. GOLDBERGER, A. s.

1964. Econometric theory, New York: John Wiley and Sons, Inc. HOEL, P. G.

1962. Introduction to mathematical statistics, New York: John Wiley and Sons, Inc. HoPKIN, J. A. and R. C. KRAMER

1965. Cattle feeding in California, Bank of America, N. T. & S. A. IsARD, WALTER

1956. Location and space economy, Massachusetts Institute of Technology and John Wiley and Sons, Inc.

JEFFERY, R. c. 1965. The logic of decision, New York: McGraw-Hill Book Co.

JOHNSTON, J. 1963. Econometric methods, New York: McGraw-Hill Book Co., Inc.

JuDGE, G. G. and T. D. WALLACE 1959. Spatial price equilibrium analysis of the livestock economy, I: Methodological develop­

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46 Bulwo'k and Logan; Cattle FeeaZot Mar'keting Decisions Unaer Uncertainty

RADNER, R. and J. MARSCHAK 1954. Decision processes (ed. R. M. Thrall, C.H. Coombs and R. L. Davis), Chapter V,

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,

1250·4,'72 (Q156) JF

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